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Why Emojis Are Now Big Business
The cartoon faces on your phone convey more emotion than most words. The founder of ad platform Emogi explains why they are serious business.
By Damon Brown@browndamon
Emojis are becoming more important than words, particularly when it comes to talking via text or social media. Twenty-four-year-old founder Travis Montaque is trying to prove it with Emogi, an advertising platform that helps brands connect with customers with emoticons.
I talked to Montaque about why emojis are some of the most emotional symbols we can use, and how his young company is using a new platform, Wink, to connect people with their favorite brands.
Inc.: When did you start your business path?
Montaque: I actually started working at Chik-fil-A at 15. I was promoted to my first leadership role, and, within a couple of years, I oversaw $7 million in revenue and hundreds of employees while helping with the company's southern expansion.
I attended the University of Miami, which isn't far from where I'm from in South Florida, and I realized that I didn't want to just help grow the company. I switched to finance and went straight into private equity investing. I was curious about big data, so, while I was studying finance and putting in 30-hour weeks at a firm, I started my first company.
We started with a curated news and video app, even though I had no background in tech. It was a lot of tinkering. Users could respond to the news posts, and we began noticing that, even if they didn't use any words, they would almost always share some kind of reaction to the story as an emoji. That's when we focused our company, Emogi, on providing 12 emoji reactions.
People began responding and reacting nonstop. It wasn't really about the content, as there were a lot of news apps even at the time, but people wanted to quickly share their feelings about something. I realized emojis were a really powerful way of expression.
I also realized that emojis are rich with data. Emogi turned its methodology toward people who want responses: consumer sentiment towards ads, giving insights into people and retooling customer engagement.
I had worked at Barclays, and then at Goldman Sachs, and then I had a great Emogi team and quit. It was a while before I could work on Emogi full time.
One of the toughest decisions is knowing when to go full time. How did you know?
I realized I was solving the right problem and I had the right team to execute it. The last point of my leaving was my raising the initial money [$200,000] from a VC. It wasn't a lot, but it gave validation that I was the right person with the right concept.
Did you always want to be an entrepreneur?
I was always entrepreneurial, from doing car washes as a kid to my later financial industry work. The big turning point was when I saw a crazy stat: Ninety percent of the world's info was created in the past two years. The first thing that rang out was information overload. I dug deeper, and saw an opportunity to [simplify communication and expression] with Emogi.
Emogi helps people express themselves, but what is the process for the brands?
Most conversations today are in characters, but we aren't there. According to our research, 50 percent of all messaging characters are emojis. Brands are missing the conversation.
Our new platform, Emogi Wink, is the first native setup for branded content: emojis, stickers, and gifs. We've partnered with messaging applications to upload brand content and help understand the context of a conversation. Along with geolocation and audience information, we leverage the data to give the right ad at the right time.
If I'm Starbucks and the messengers are talking about being tired, I can offer a coffee discount via the Wink platform. The content is triggered on any standard Unicode [mobile] keyboard, so the consumers can just keep using their favorite keyboard.
Emogi wants to help brands join the conversation, not interrupt it.
Emojis have been popular for years, especially if you look at eastern Asia. Why is this becoming a serious conversation now?
Emojis are really being driven by mobile. A big uptick happened in 2011 when Apple introduced an emoji keyboard. It was a huge tipping point!
A softer reason is that it helps people be understood by building a personal connection. Back in the day, when people talked, they did it in person and could observe facial expressions and voice tone. With the rise of digital communication, the ability to understand sarcasm and other things is harder. Emojis are introducing something back when it comes to connecting with people in a digital world.
You're 24 and have been a founder for years. How are you balancing the intense business pressure with the self-exploration that usually happens in your 20s?
Those things aren't mutually exclusive. When people have to do something difficult or are thrown into different environments, it only accelerates the way you find yourself. I remember realizing I couldn't get to where I wanted to be living in Miami, so seven days later I moved to New York. My decision to jump into entrepreneurship headfirst helped me quickly discover who I am and what I stand for. Those beliefs formed me into who I am today.
As entrepreneurs, it is rarely about the product, but the effect the product could have on others. What impact do you want Emogi to make on the world?
There is a notion that ads on mobile or online platforms are a waste of time. You have ad blockers on the rise and increasingly vocal consumer complaints. That makes me sad. When you look at the Super Bowl, you see people gathered around the television actively seeking out commercials. Ads are creative. And, if they are done effectively, people will enjoy them.
I want to bring an element of fun and joy back into mobile. Let's help brands get their message across and give great experiences to the consumer.


The Agile Alliance Functional Testing Tools Workshop (AAFTT), held the day before the Agile 2011 conference in Salt Lake City, was once again one of the highlights of the conference. Organised by Jennita Andrea and Elisabeth Hendrickson, it was as always a wide variety of participants with a passion for testing and testing tools. Here are my notes from the day held on August 7, 2011.
The session was facilitated by Ainsley Nies, and all of the official session notes are stored on the AAFTT wiki: http://aaftt.agilealliance.org:8080/display/AAFTT/agile2011.
The theme of the AAFTT is: "Advancing the state of the art and the practice of Acceptance Test Driven Development".
The law of mobility and responsibility (also known as the law of two feet) is if you are not learning or contributing where you are, go some place where you will. Also, butterflies and bumblebees cross pollinate ideas.
Finally, we were warned to be prepared to be surprised.
This was the first session that I attended.
Elisabeth Hendricksen led this session on tools that are attempting to solve the problem at the last mile.
Mary Gorman led this discussion.
As with any conference, there are always sessions you would have liked to have got along to.
Richard Lawrence led a discussion on Static Analysis for Gherkin which turned into a discussion on design patterns for Cucumber.
My mate Jason Montague led a session on Building Conditions Conducive for ATDD Adoption.
As for me, my comment was the day had renewed my energy again. ATDD is hard, and as a community we need to try harder.
Finally, I recorded a short audio podcast for The Agile Revolution wrapping up AAFTT.
Small update to include link to podcast.


Does anyone have the same problem ? What is going wrong there ?
It appears you're trying to install kdelibs3-devel-3.0.5-28 and kdebase3-3.0.4-76; if so, that is incorrect. Note the version numbers do not match: 3.0.5 versus 3.0.4. Make sure all your kde rpms come from the same "version family".


1. On a PC, the video wall process for system user a exists while the C/S is not started.
2. System user b remotely logs in to the PC and forces system user a to log out of the operating system.
3. System user b starts the C/S and clicks Log Out.
4. During logout, the C/S recognizes the video wall process for system user a and asks system user b whether to stop the video wall process.
5. The process is not stopped when system user b clicks OK.
Multiple CUs or video wall processes cannot be enabled concurrently on the same PC.
Step 1 On the PC where the eSpace IVS C/S client is installed, start the task manager, click Users, and check whether the operating system is used by multiple users.
 If the operating system is not used by multiple users, contact Huawei technical support engineers.
 If the operating system is used by multiple users, contact these users and ask them to log out. Then go to the next step.
Step 2 Restart the eSpace IVS C/S client and check whether the fault is rectified.
The eSpace IVS C/S client does not differentiate system users when identifying video wall processes. In addition, different operating system users of the PC where the eSpace IVS C/S client is installed cannot share resources.


Fearless Hyena is an 80's kung fu sequel following a pair of evil gung-fu artists, Heaven and Earth, are slaughtering the entire Yin-Yang brotherhood. he movie opens with two members of the brotherhood and their two male children being chased by Heaven, Earth, and various bad guys. The two members get away separately with their children. Twenty years later we see Jackie Chan as he hunts frogs and snakes and keeps them in his pants.


I am sure that you will agree that the after sales services or the customer care in India is one industry which needs to be addressed at priority. I recently had a very bad experience with one of the leading telecommunication company. And this was not the first time. The leading brands in almost every field - electronics, retail, services, gadgets - fail at providing the customer services which can be even called "average".
Of lately, I had to restore to reaching the brands which harass the customers using all the means - telephones, emails and their social media network. I realized that probably just speaking personally to the customer care executive doesn't help. They are just employees. But probably publicly speaking about them on the social media makes them react faster. Because NOW they know that the brand image might get affected - largely.
I am in no mood to bash any particular brand. I rather did a li'l bit of research(given the nature of my job which helps me pay my bills!) that why brands need to be vigilant on the social media - specially when the customer is so social media savvy!
Social media is the number one way that most of us stay connected, stay informed and interact with our world, in 2017. In the USA, 7 in 10 people use social media daily and worldwide, social media is now used by nearly 70% of the world's population. According to statistics and facts from Websitebuilder UK, 90% of social media users have used it to communicate with brands. Social media is now the most preferred way customers want to be interacted with for their customer care needs.
95% of businesses indicate their customers care has a direct impact on their brand image, and good customer care boosts a businesses customer loyalty, brand advocacy, referrals, ROI, sales, and revenue.
With over one-third of customers preferring social media as their primary channel of customer care and 63% of customers expecting companies to provide customer service via social media, the market has shifted from the conventional phone/dated operator based call centers.
Though the customers are clear on their preferences, many brands are still not meeting their expectations. 80% of companies say they deliver exceptional SMCC, but only 8% of their customers agree. The main concern in the sector is that 95% of customer complaints on social media don't reach companies. Furthermore, only 12.9% of the informational demand is met with a response from brands and one-third of customer complaints are ignored altogether.
Another major discrepancy in customer expectation and brand delivery is in the average answer speed through social media. Currently, 42% of customers who lodge a complaint through social media expect a response within 1 hour. A further 32% expect a response within 30 minutes and 11% expect instant support. The average response time, currently at over 5 hours, simply is not acceptable to the customer and needs to be addressed.
With so much riding on a brand's SMCC strategy and the industry simply under-delivering in terms of expectation, many brands are losing big. Poor response time can lead to a 15% increase in customer churn, and 31% of people post online after a bad customer care experience. This exponentially affects the brand's image and is a detriment to the overall goal of converting and keeping customers. Additionally, 30% of people will go to a competitor if a brand doesn't respond decreases customer advocacy by as much as 50%.
On the flip side of that equation, good SMCC can improve customer satisfaction by 26.6% and 71% of consumers with positive SMCC experiences are likely to recommend the brand to others. 75% of people share good experiences with brands on social media, and 65% of people have more brand loyalty when a brand reaches out to social media.
With SMCC being the number one preference for customer care needs, brands need to ensure dedicated support via the channel. Make sure messages are answered promptly and build strong communicative relationships with your consumers on social media. A lack of vigilance in SMCC will only result in losses of both customer loyalty and profits.


Diabetes is associated with an increased risk for a number of serious, sometimes life-threatening complications, and certain populations experience an even greater threat. Good diabetes management can help reduce your risk; however, many people are not even aware that they have diabetes until they develop one of its complications.
African-Americans are significantly more likely to suffer from blindness, kidney disease and amputations.
For these reasons, the American Diabetes Association has prioritized the elimination of disparities by placing disparities prominently in our 2011-2015 strategic plan and 2014 legislative priorities.


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Mira Loma High School
COVID pandemic increases mental health needs on high school campus
Kayli Huang, Mira Loma High School
This year, a Peer Mediation Group was made available to students at Mira Loma High School to support mental wellness. This group of students help their peers resolve minor conflicts and problems and steer them to resources for larger issues.
The program is launching in the wake of a nationwide spike in youth mental illness.
The COVID-19 pandemic has brought youth mental health into the spotlight as many children and teens have found themselves isolated from friends and dealing with greater stress ‒ especially during periods when schools have gone remote.
The number of minors struggling with mental health has been climbing, according to Mental Health America (MHA) National, a nonprofit dedicated to promoting mental health. In 2021, 206,000 more youths (age 12-17) have experienced major depressive episodes in the United States compared to 2020. Most of these young people ‒ 59.6 percent ‒ do not receive any mental health treatment, according to MHA.
According to the National Alliance on Mental Illness (NAMI), the largest grassroots mental health organization in the US, 50 percent of all lifetime mental illness begins by age 14, and 75 percent by age 24. In other words, a lot of people who experience mental health challenges are likely to experience them before finishing high school.
Aaima Arshad is a junior at Mira Loma High School and a peer mediator. She explained that in order to become a peer mediator, training was mandatory.
"During school, we had to do mock mediations in groups and were put in the positions of both the mediator and the person having the conflict," Arshad says, "This helped us understand what it means to be in the role of a mediator and what the person with the conflict is going through, while also helping us gain experience in being a mediator. But before all of this, we were taught the process of mediation and were shown several videos of mock mediations."
While mock mediations are a great way to understand the process of peer mediation, Prachiti Kamle, a junior at Mira Loma High School and peer mediator, described the meaning behind the style of training.
"We had a 2 day training in which we covered a variety of topics to help us better help students who come to mediation," Kamle says, "A lot of our training was focused on understanding why people may feel a certain way, and how to help them convey these emotions and feelings in a respectful yet impactful manner, since sometimes the conflict could just be a misunderstanding between peers. After that, we specifically focused on methods of reaching a compromise or mutual agreement, hopefully helping both sides see why the other feels the way they did, and what they both can do in the future to resolve the issue."
There are many reasons for why high school students would want to take on the role of being a peer mediator to help out other students in their school community. For Arshad, it was the ability to gain experience in the medical field while giving back to her community.
"I feel like it is a great way of helping others and getting back to the community. I also possibly want to be a psychiatrist in the future, so counseling or being a help to others let's me try out some aspect of counseling," Arshad says, "I generally like the idea of being able to be there for others in times of need and being able to resolve conflicts would act as a way to do that. I also think that things as such help you gain experience in not only mediation and being there for others, but also being able to solve your own conflicts because at the end of the day, we all have problems."
According to Kamle, another motivator for students to take on the role of peer mediator is the opportunity to improve the well being of the school as a whole by helping one student at a time.
"The most valuable thing to me when I am involved in a conflict is when someone is willing to listen (and even offer advice on occasion), as it keeps me from spiraling further into the situation," Kamle says, "I wanted to be a similar 'listening ear' for students at Mira Loma, and helping others become more open-minded makes for a more compassionate and caring school environment in my opinion."
As this program is starting up, there is lots to be done, especially as the whole world moves out of the pandemic and being socially cut off from everyone else, Arshad states.
"We have not done any actual mediation so far but from what I would assume, people are a lot more distant and would have trouble socializing again."
There are also some benefits that have come out of the pandemic, Kamle explains. Isolation, in a way, allowed for some other bonds to be strengthened.
"I think the isolation aspect of the pandemic took a major toll on the mental health of my peers and I. It was definitely difficult going from seeing each other every day to barely at all, and many of us are still working on rebuilding that social aspect of our lives," Kamle says, "These dramatic life changes were definitely not easy for many, especially when we had so much thrown at us in such a short period of time. However, I also know that many people relied on their friends for emotional support even more during the pandemic, creating stronger bonds between them."
According to Kamle, "Although the isolation was difficult, it allowed for many to reflect on their mental health, and may have provided more opportunities to process emotions and heal."
For students at school, councilors are always available to meet whenever a problem arises. There are also several crisis hotlines like The Source by the Sacramento Children's Home, the Crisis Text Line, and the National Suicide Prevention Lifeline for youths to get help, or when they just need someone to talk to.
Kayli Huang, Reporter
I'm Kayli Huang, a junior at Mira Loma High School. I love being part of the Student Education Reporter Program as it has given me an outlet to communicate...
Technology not a cure all say both students and teachers
Pleasant Grove ponders opposing views on switch to lock schedule
Horizon Charter School
Law enforcement agencies collaborate to present District Attorney's Youth Academy to local teens
Standardized tests increasingly controversial at secondary level as colleges drop required tests
Students say winter final exams result in increased stress levels
Are later secondary school start times helping or hurting California high school students?
Sacramento area youth volunteer to help facilitate midterm election
UC, community colleges create new transfer opportunities for California students
St. Francis pioneering ethnic studies course for Catholic schools

redpajama_set_name: RedPajamaCommonCrawl


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Tag Archives for " Gollancz "
EP204: One Page Punch-Ups SF&F special
In the latest of our One Page Punch-Up specials we asked our Patreon Chart Topper supporters to send in one page from their work-in-progress and we gave them Gollancz's Gillian Redfearn and Marcus Gipps to review and critique. Between them Gillian and Marcus have worked with some legendary authors, including Joe Hill, Joanne Harris, Joe […]
EP198: Lorna Cook – A Bestseller Dream Come True
Lorna Cook is yet another listener who used what she had learned on the podcast to launch her career and in special episode we speak to her before her novel The Forgotten Village was published, and we discover how she became the UK's top selling Kindle title! https://www.patreon.com/bestsellerexperiment Our novel Back To Reality is out […]
EP196: Kiley Dunbar – Another Listener Who Got A Deal!
These are our favourite episodes where we speak to one of our listeners who got a book deal! Kiley Dunbar is the author of One Summer's Night, the first of a three book deal with Hera Books. Kiley tells us the role that the podcast and the RNA Writers' Scheme played in getting her debut […]
EP179: Chris Humphreys – From The Stage To Page
Chris Humphries started as an actor but his writing career took off when he took one of his key roles and created an acclaimed series of novels.
EP173: Gavin G Smith – Rock n Roll Book Promotion
Gavin G Smith is the author of The Veteran and The Bastard Legion science fiction series. When it came to promoting his latest book, he decided that a few Tweets simply wasn't enough. Instead he chose to get the band together and produce a music video and song that ties-in to the series… To support […]
EP150: Joanna Nadin – The Queen of Bloody Everything
Joanna Nadin has written over 70 award-winning books for children and adults and recently inspired Mr. Stay to completely re-write his middle-grade children's book. She shares a flood of amazing writing tips with us and tells us why she loves to make readers cry. To support the show, check out bestsellerexperiment.com/support Our novel Back To […]
140: The Conventions & Comic Cons Survival Guide
In these highlights from our YouTube Live show with Gollancz's Marcus Gipps and Stevie Finnegan we look into the world of conventions, comic cons and other bookish events. AND we have some exclusive news about GollanczFest! To support the show, check out bestsellerexperiment.com/support Our novel Back To Reality is out now bestsellerexperiment.com/backtoreality PODCAST […]
EP128: Youtube LIVE show with Ed Wilson (Mr. Stay's agent spills the beans on money and more)
This is not the usual 'How do you get an agent?' show, instead we talk about money, author's earnings, shrinking advances, publishers' profits, and that letter from the Society of Authors to the Publishing Association (and we get sidetracked into a conversation about socks)… To support the show, check out bestsellerexperiment.com/support Our novel Back […]
EP123: Warren Adler – The Master of Dysfunction
Warren Adler is best known for his classic bestseller The War of the Roses and has had the most incredible career as a writer of novels, stage plays, poems, essays and short stories. He was also an eBook pioneer and paved the way for countless indie authors. Warren's first novel was published when he was […]
EP120: The Quanderhorn Xperimentations with Rob Grant and Andrew Marshall
Mr. Stay spoke to a pair of his TV writing heroes at the MCM Comic Con in London recently. Rob Grant is the co-creator of the classic science fiction comedy Red Dwarf, and Andrew Marshall is best-known for the BBC comedy Two Point Four Children. Together they have created a fantastic BBC radio comedy series […]


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Slot: 29D-2 Dec. 29, 3:45 p.m. – 5:45 p.m.
Panel: Modernist Travel through Imperial Spaces
Chair: Evgenii Bershtein, Reed College
Title: "Gogol in Palestine": Toska and Obnazhenie in Viktor Shklovsky's
A Sentimental Journey
Author: Anne Dwyer, University of California, Berkeley
As Viktor Shklovsky's A Sentimental Journey (1923) follows its hero's peregrinations from Petrograd to Galicia, Persia, the Caucasus, and Ukraine, the wartime dissolution of boundaries comes to the narrative's fore. Armies, political borders and landscapes are repeatedly bared (obnazheny), meaning that they are both literally abandoned and figuratively revealed to be arbitrary and conventional. On a metafictive level the memoir reads as an illustration of the formalist concept obnazhenie priema (baring of the device) as it draws attention to its own montage-like construction.
This presentation examines an intertextual moment in which the wandering narrator likens his own misery (toska) to Gogol's experience in Palestine. The reference is to a late letter from Gogol to Zhukovsky that stands out for its quasi-formalist language: Gogol describes the barren Palestinian landscape as strannyi (strange) and obnazhennyi (bare), repeating the latter term frequently.
Shklovsky's dialogue with Gogol not only reveals a Russian source of the term obnazhenie priema, but also sheds light on central questions of Shklovsky's literary and theoretical oeuvre. Gogol and Shklovsky both address the risks of traveling to exotic locales for cheap artistic effect; both seem to take a perverse pleasure in finding the bared ravages of history at the ends of their journeys; both also associate the process of obnazhenie with nomadism. Paradoxically, Gogol's repeated use of the word obnazhennyi in the ideologically charged context of a pilgrimage to the Holy Land adds cultural and historical depth to a term that ostensibly seeks to free art from the baggage of motivation and ideology. This paper suggests that by referring to Gogol's pilgrimage, A Sentimental Journey bares its own status as a modernist pilgrimage to the Holy Land of obnazhenie—the war-torn borderlands where war, history and art reveal themselves in all their nakedness. Moreover, Shklovsky's passing identification with Gogol questions the possibility of a naked, defamiliarized view of historical and literary objects while still affirming the search for a defamiliarizing perspective and innovative artistic practice.
Title: The Archaeology of Russian Modernist Scythianism
Author: Michael Kunichika, University of California, Berkeley
This paper examines how the classical and primitivistic impulses of Russian modernism found their primary locus in the Steppe and in the regions surrounding the Black Sea. The region appeared to Russian modernists as a quintessential borderland, revealing strata both mythic and historical. Once the edge of the known world separating the ancient Greeks from the Scythians, it became, millennia later, the periphery of the Russian empire. Emblematic of the modernist period's ability to connect archeology and aesthetics, the Black Sea region presented Russian modernism with an exemplary site upon which the primitive and the classical converged in a crucial juxtaposition, and served to delineate the leading schools of Russian modernist art and literature. This paper begins by focusing on the pronounced orientation towards Hellenistic Athens and Scythian "barbarians" in the work of Viacheslav Ivanov, Osip Mandel'shtam, the archeologist and historian Michael Rostovtseff, Benedikt Livshits and Velimir Khlebnikov.
Insofar as this paper will claim a more pronounced relationship between archeology and aesthetics, I examine, in parallel with these writers, the work of Michael Rostovtseff, an archeologist and later a famous historian of the Roman empire, who was also a close member of Ivanov's circle The Tower. Rostovtseff's Iranians and Greeks in the South of Russia (1918) provided material evidence to draw the Scythians, and hence Russia, into the cultural orbit of antiquity, and I argue, supplied these poets with the archeological underpinnings of their aesthetic programs. Set against the Hellenistic impulse of Russian modernism, this paper will focus principally on the Scythian theme of Russian modernism. It puts forward the interpretation that the Scythians, and well as other "Asiatic nomads" supplied Russian modernists with an ostensibly indigenous ancestor upon which to base their own experience of modernity as exile or displacement.
Title: Japan and Velimir Khlebnikov's Eurasia
Author: Susanna Lim, Wheaton College
This paper will discuss the significance of Japan in the work of the futurist poet Velimir Khlebnikov (1885-1922). The poet's wish to seek out the "laws of time" governing history was first prompted by his reflections on the Russo-Japanese War (1904-1905). Khlebnikov was also deeply interested in Japanese language and art: he was fascinated by ukiyo-e, and, in a letter to A. Kruchenykh (1913), discussed the possibility of applying aspects of Japanese language and verse to Russian literature.
Khlebnikov's attraction to Japan can be understood first and foremost in relation to the admiration and sympathy for the East prominent in his work. However, in the context of the poet's vision of a Eurasian utopia, Japan's place is not entirely unproblematic.
In this presentation I examine two of Khlebnikov's works related to Japan, "Letter to Two Japanese" (1916) and the poem "The Coup d'État in Vladivostok," (1922) in their historical context. The first is the Futurist's enraptured response to letters written by two Japanese youths to Russians and published in the Japanese daily Kokumin Shimbun and Russkoe Slovo; the second, one of Khlebnikov's last works, depicts the Japanese occupation of the Russian Far East during the Civil War.
I suggest that although reflecting an ideal of a harmony between East and West, these two texts at the same time also point to a disjuncture between the poet's utopianism, on the one hand, and the reality of geopolitics and imperial maneuvering, on the other, a disjuncture that is translated into ambivalence about Japan's place in Eurasia.
At issue is the shifting place of Japan in Khlebnikov's Eurasia: the poet's desire to create a Eurasian utopia free from the fetters of the West comes to stand in increasing tension with the reality of Japan's imperial ventures and its occupation of the Russian East.


text:
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redpajama_set_name: RedPajamaGithub


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Home Academy of Accounting Historians 1
Accounting records of Quakers of West Falmouth, Massachusetts (1796-1860): An analysis
The Accounting Historians Journal Vol. 18, No. 2 December 1991
Jayne Fuglister CLEVELAND STATE UNIVERSITY and Robert Bloom JOHN CARROLL UNIVERSITY
Abstract: The village of West Falmouth, Massachusetts was settled in the 1660s by William Gifford and other Quakers who came there to avoid persecution. They lived relatively isolated from other settlers in the region. The accounting records of Prince Gifford, Jr. (1771-1853) and Prince Gifford Moore (1812-1885), descendants of William Gifford, are still in existence. This paper provides an analysis of these records, which reflect the simplicity, frugality, honesty, and equality of early West Falmouth Quakers. Littleton's antecedents of double-entry bookkeeping are applied to explain the use of the single-entry system of accounting by West Falmouth Quakers during the same period that Philadelphia Quakers were using the double-entry system.
This study examines accounting records of early American Quakers in West Falmouth, Massachusetts and analyzes the records within the context of the environmental conditions pre-vailing at the time the records were prepared. West Falmouth Quakers were semi-isolated compared to the Pennsylvania Quak-ers;1 and, accordingly, the records of the West Falmouth Quakers are considered likely to reflect the religious customs of Quakers in Colonial America. Additionally, West Falmouth Quaker account-
The authors are grateful for the assistance of Cecelia Lucinda Bowerman, and Harriet Quimby and Hannah Fitts of the Falmouth Historical Society for providing the records analyzed in this paper. Additionally, the authors thank two anonymous referees and Joe McKeon and Marilynn Collins for comments on this paper.
1Evidence that West Falmouth was settled mostly by Quakers, who were thus semi-isolated, is provided in Deyo [1890]:
"In 1678 lands were laid out at Oyster pond; also at Hog Island and great
Sipperwisset where the early settlers were William Gifford, Senior; Wil-
Title Accounting records of Quakers of West Falmouth, Massachusetts (1796-1860): An analysis
Author Fuglister, Jayne
Subject Quakers -- Massachusetts -- West Falmouth
Church finance -- Accounting
Account books -- History
Geographic Location West Falmouth (Mass.)
Abstract The village of West Falmouth, Massachusetts was settled in the 1660s by William Gifford and other Quakers who came there to avoid persecution. They lived relatively isolated from other settlers in the region. The accounting records of Prince Gifford, Jr. (1771-1853) and Prince Gifford Moore (1812-1885), descendants of William Gifford, are still in existence. This paper provides an analysis of these records, which reflect the simplicity, frugality, honesty, and equality of early West Falmouth Quakers. Littleton's antecedents of double-entry bookkeeping are applied to explain the use of the single-entry system of accounting by West Falmouth Quakers during the same period that Philadelphia Quakers were using the double-entry system.
Citation Accounting Historians Journal, 1991, Vol. 18, no. 2, pp. 133-154
Date-Issued 1991
Source Originally published by: Academy of Accounting Historians
Rights Copyright held by: Academy of Accounting Historians
Digital Publisher University of Mississippi Library. Accounting Collection
Date-Digitally Created 2005
Identifier ahj18-2-1991 p133-154
Description-Abstract The Accounting Historians Journal Vol. 18, No. 2 December 1991 Jayne Fuglister CLEVELAND STATE UNIVERSITY and Robert Bloom JOHN CARROLL UNIVERSITY ACCOUNTING RECORDS OF QUAKERS OF WEST FALMOUTH, MASSACHUSETTS (1796-1860): AN ANALYSIS Abstract: The village of West Falmouth, Massachusetts was settled in the 1660s by William Gifford and other Quakers who came there to avoid persecution. They lived relatively isolated from other settlers in the region. The accounting records of Prince Gifford, Jr. (1771-1853) and Prince Gifford Moore (1812-1885), descendants of William Gifford, are still in existence. This paper provides an analysis of these records, which reflect the simplicity, frugality, honesty, and equality of early West Falmouth Quakers. Littleton's antecedents of double-entry bookkeeping are applied to explain the use of the single-entry system of accounting by West Falmouth Quakers during the same period that Philadelphia Quakers were using the double-entry system. This study examines accounting records of early American Quakers in West Falmouth, Massachusetts and analyzes the records within the context of the environmental conditions pre-vailing at the time the records were prepared. West Falmouth Quakers were semi-isolated compared to the Pennsylvania Quak-ers;1 and, accordingly, the records of the West Falmouth Quakers are considered likely to reflect the religious customs of Quakers in Colonial America. Additionally, West Falmouth Quaker account- The authors are grateful for the assistance of Cecelia Lucinda Bowerman, and Harriet Quimby and Hannah Fitts of the Falmouth Historical Society for providing the records analyzed in this paper. Additionally, the authors thank two anonymous referees and Joe McKeon and Marilynn Collins for comments on this paper. 1Evidence that West Falmouth was settled mostly by Quakers, who were thus semi-isolated, is provided in Deyo [1890]: "In 1678 lands were laid out at Oyster pond; also at Hog Island and great Sipperwisset where the early settlers were William Gifford, Senior; Wil-
Accounting records of Quakers of West Falmouth,...


You're invited to come play!
Join us at one of our many community events, such as the annual Bike MS: Waves to Wine Ride or Press Democrat Food and Wine Festival. Enjoy an outdoor SOMO Concert or sit-down to a farm-to-table meal or indoor event at Sally Tomatoes. Stop by at lunchtime for Food Truck Tuesday/Thursday between 11:30a-1p or take a self-guided tour of the latest Art installations. You're welcome anytime!
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North Coast Wine & Food Festival by Press Democrat at SOMO Village!


Last week I had the chance to get into the photostudio at JamFactory, where I am currently resident artist, to take some photos of a new collection of Archaeologic vases.
More images are now uploaded, here.
Posted on Tuesday, May 12th, 2015 at 1:03 am in Uncategorized | | Both comments and pings are currently closed.


An awkward, curious girl growing up in a foreign country, Stephanie LaCava finds solace and security in strange and beautiful objects. When her father's mysterious job transports her and her family to the Parisian suburb of Le Vesinet, the young American embarks on a life of discovery. Tasting the enchantments of Paris, she makes friends with her peers at a wildly unconventional international school and faces terrorism. But Stephanie's wonder gives way to anxiety and a deep depression brought on by a series of circumstances.
Through her darkest moments, Stephanie continues to filter the world through her peculiar lens, discovering the strange beauty surrounding her. The grotesque (beetles and catacombs), the natural (mushrooms and lilies of the valley), the cultural (Nirvana and other nineties touchstones), and the historical (Nancy Cunard, Jean Seberg), all become unique talismans. Encouraged by her father through trips to museums and scavenger hunts in antique shows, she traces an interconnected web of stories of past outsiders, historical and natural objects, and her present predicament that ultimately helps her survive when she feels she's losing control.


At Moco Solutions we fix all different types of mobiles and screens.
not listed then please email or contact Moco Solutions for a quote.
You will need just a Front Screen Digitizer if your Front Screen is cracked or the touch screen has stopped working and your LCD is still fully working.
You will just need a LCD Screen if the actual LCD screen has stopped working or is damaged but the front screen is not cracked and the touch screen is fully working.


This Siberian Husky nicknamed Taz sold and went to a new family.
This loving Siberian Husky puppy will make a great addition to your family. He is friendly and well socialized. This puppy is ACA registered, vet checked, vaccinated, wormed and comes with a 1 year genetic health guarantee. Please contact us for more information or check out our website at www.keystonepuppies.com for pictures of all of the available puppies for sale.


Teaching. Eating. Singing. Eating. Playing. Eating. Ministering.
Do you see a pattern here? So was our second weekend with a group of students from Word of Life Bible Institute, Mexico.
This weekend Karen Méndez joined us, and Lord willing she'll be able to continue with Matheus, Cheila and Rebeca.
Our very first kids club at our current location had 11 kids in attendance, with a lot of potential for growth. Saturday afternoon was spent celebrating an anniversary at a nearby rehab center that we try to support, and ended with a meal of mole poblano and rice. Hmm. Youth group met at our house for the second week, with some new kids in attendance.
After youth group on Saturday, the Cottrills, the Word of Life crew and us had a planning time, brainstorming and dreaming about upcoming events, and what we would like to see in the new church plant. What a fun time! Keep in mind as you try to read the picture that I was writing side-ways and upside down half of the time!
We finished out the day eating alitas, or chicken wings, at a local restaurant. We'll not go into a whole lot of detail as to how several of us spent the wee hours of the morning with all that mole and hot sauce in our stomachs!
On Sunday, Rebeca and Karen taught the young people and Matheus and Cheila sung a closing number at the Jesús María church. Great times! Oh, then we finished out the weekend with some marinated chicken breast, pasta, salad and…wait for it…ice cream.


must be ready dissertation proposal project management to turn sharply from our present ways. The film passed lightly over the history of Vietnam and assured viewers of Americas good intentions but nonetheless revealed how routine violence employed by the arvn and its American advisers was turning the population against the government. . During the eleven-day onslaught, more than 36,000 tons of munitions rained down on North Vietnam, resulting in at least 2,200 civilians killed. . The trees had literally grown to death because thats how Agent Orange works it accelerates growth in a plants cell until finally the plant or tree dies. .
Socialist Party chairperson Michael Harrington, however, was a strong advocate of political action, even arguing that the peace movements resources would be better spent on influencing Congress than on organizing mass demonstrations. 374 David Farber, Chicago 68 (Chicago: University of Chicago Press, 1968. Americans reacted with shock and confusion. . Those who were apprehended, however, were rarely NLF leaders. . Saigon, April 30, 1975 (Agence France Presse) On April 20, 1975,.S. Jack Valenti, special assistant to the president, told Johnson in April, We simply arent doing our propaganda job right in this country. . A Gallup poll taken in June 1965 reported that 66 favored continued.S. 341 The issue eventually reached the Supreme Court which ruled unanimously in favor of Bond. A boatload of about 23 Jews sailed into the remote Dutch port of New Amsterdam and requested permission to remain.


Project Russia: Is it coming home to England?
England advanced to the World Cup quarterfinals on penalties after a 1-1 draw with Colombia on a dramatic evening in Moscow on Tuesday.
120' Penalty Shootout ends, Colombia 1(3), England 1(4).
120' Goal! Colombia 1(3), England 1(4). Eric Dier (England) converts the penalty with a right footed shot to the bottom left corner.
120' Penalty saved! Carlos Bacca (Colombia) fails to capitalise on this great opportunity, right footed shot saved in the centre of the goal.


text:
Home Gadgets Mobile Xiaomi Set to Launch the Mi 10i Smartphone with the Snapdragon 750G...
Xiaomi Set to Launch the Mi 10i Smartphone with the Snapdragon 750G Chipset
After the official launch of the Xiaomi Mi 10, the Chinese brand has been unveiling other new smartphones belonging to the Mi 10 series.
Recently, a few reports have revealed that the firm may announce a new smartphone under the Mi 10 series.
The reports further mention that the handset, which may arrive as Xiaomi Mi 10i, will launch in India soon.
Though the name of the phone is new, speculations believe that the phone may arrive as a revamp of the Redmi Note 9 Pro 5G that debuted in China last month.
Interestingly, the upcoming Xiaomi Mi 10i surfaced on Geekbench with the model number M2007J17I. Its appearance on the benchmarking platform suggests that the company may launch the device soon, hence the tests.
The name ''Xiaomi Mi 10i'' was not stated explicitly in the Geekbench listing, but the model number M2007J17I was present in the publication.
The phone is codenamed ''gauguinpro'', which shows that it is the Indian variant of the Redmi Note 9 Pro 5G.
Most likely, the Snapdragon 750G chipset with a base frequency of 1.80GHz, would drive the upcoming Mi 10 series phone.
The phone will arrive with the Android 10 OS, and the chipset will come with 8GB of RAM.
In the single-core and multi-core test, the Xiaomi Mi 10i gathered 652 points and 2004 points respectively.
Meanwhile, sometime last week, a report stated that the handset would hit the Indian market in variants like 6GB RAM + 128GB storage and 8GB RAM + 128GB storage.
The Xiaomi Mi 10i may arrive with a 6.67-inch Full HD+ IPS LCD panel that has a high refresh rate of 120Hz.
It may feature a 16-megapixel front-facing camera and a 108-megapixel quad-camera setup.
Previous articleGionee K30 Pro Arrives in China with a 6.53-inch Screen, up to 8GB of RAM, and a 4,000mAh Battery
Next articleHonor Getting Closer to Securing a Deal with Chipset Maker, Qualcomm


text:
Home»Featured»Founder of Black Lives Matter resigns after learning "ugly truth": BLM is not about rebuilding black families
Founder of Black Lives Matter resigns after learning "ugly truth": BLM is not about rebuilding black families
Herland Report: Founder of Black Lives Matter resigns after learning "ugly truth": Rashard Turner, the founder of a Black Lives Matter chapter in St. Paul, Minnesota, revealed in a video published last week that he quit the organization after he "learned the ugly truth" while being an insider in the far-left group, reports Daily Wire:
"I was a founder of Black Lives Matter in St. Paul. I believed the organization stood for exactly what the name implies, black lives do matter.
However, after a year on the inside, I learned they had little concern for rebuilding black families, and they cared even less about improving the quality of education for students in Minneapolis. I learned the ugly truth. I resigned from Black Lives Matter after a year and a half."
Turner made the remarks in a video published by TakeCharge, a black-led organization that rejects critical race theory and woke culture.
Founder of Black Lives Matter resigns after learning "ugly truth": In the video, Turner says in part:
I was born in Minneapolis in 1985. We called the north side home at that time, 18th and Queen. When I was two years old, my father was shot and killed. My mother wasn't able to take care of me. So I was raised by my grandparents. They told me that if I was going to change my life for the better, education was the answer.
So I worked hard in school, I got into Hamlin University and earned a college degree, first in my family.
Marxist Black Lives Matter Patrisse Khan-Cullors divisive Race Ideology serves Interests of Ultra Rich
Almost 100 % of riots destroying property is done by Black Lives Matter, study shows
Marxist Black Lives Matter leader push to end private property rights
Marxist Black Lives Matter says it is anti-democracy
Founder of Black Lives Matter resigns after learning "ugly truth": Turner: Then I went on to earn a master's in education from St. Mary's University of Minnesota. I am living proof that no matter your start life, quality education is a pathway to success. I want the same success for our children in our communities.
That's why in 2015, I was a founder of Black Lives Matter in St. Paul. I believed the organization stood for exactly what the name implies, black lives do matter.
However, after a year on the inside, I learned they had little concern for rebuilding black families, and they cared even less about improving the quality of education for students in Minneapolis.
That was made clear when they publicly denounced charter schools alongside the teachers union. I was an insider in Black Lives Matter. And I learned the ugly truth.
The moratorium on charter schools does not support rebuilding the black family. But it does create barriers to a better education for black children. I resigned from Black Lives Matter after a year and a half. But I didn't quit working to improve black lives and access to a great education.
Watch the YouTube video here.
The level of censorship in social media and search engines is all-time high. Do like thousands of others, subscribe to The Herland Report newsletter here! Led by Scandinavian bestselling author, Hanne Nabintu Herland, The Herland Report news and opinion website provides independent analysis from leading Western intellectuals and ground breaking YouTube interviews, cutting through the mainstream media rhetoric. It is a great place to watch interviews and read the articles of leading intellectuals, thought leaders, authors and activists from across the political spectrum. The Herland Report believes in freedom of speech and its editorial policy resides above the traditional Left vs Right paradigm which we believe has lost its relevance and ability to describe the current driving forces in Western politics.
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Neil Patrick Murray passed away on March 5, 2019, at the age of 93 in Cottonwood, Arizona. He was born on January 24, 1926, near Union, Oklahoma to Herbert Dale and Florence Zimmerman Murray. He attended schools at Verden, Dover, Mustang, and Putnam City. As a boy he began working in construction helping his father build homes in Oklahoma City. He was drafted into World War II while a senior in high school, trained at Birmingham, Alabama, and then sent to Germany via Scotland and England. He served 3 1/2 years as a mechanic in the army overseas. After his discharge he returned to Oklahoma City and resumed working in construction. Neil was employed for 35 years with J. W. Skaggs Construction Company as Job Superintendent. Neil was an avid beekeeper for many years, a fanatic fisherman, and a serious gardener. He is survived by his wife of 68 years, Ruth Ann, and 3 children, 6 grandchildren, and 8 great-grandchildren. He is also survived by his brother Bob Murray of Kingfisher, Oklahoma. A memorial will be held at a later date.
To send flowers or a remembrance gift to the family of Neil Murray, please visit our Tribute Store.


text:
Anglican priest David Norton pleads guilty to sexual abuse of boy
CTV London Published Wednesday, February 28, 2018 1:51PM EST Last Updated Wednesday, February 28, 2018 6:59PM EST
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A former Anglican priest pleaded guilty to sexual touching involving a boy under the age of 14 in a London court on Wednesday from incidents in the 1990s.
David Norton still faces further charges from alleged incidents from the 1970s when he was a priest at St. Andrew's Anglican Church on the Chippewa of the Thames First Nation reserve near London. He was also lecturing at King's University College.
Police said an investigation has found allegations of abuse involving First Nation boys, starting in 1977.
The Diocese of Huron removed Norton's permit to act as an Anglican priest until the conclusion of the trial.
In a statement from the time of the charges in 2015, King's University College said Norton was placed on a leave of absence, pending the outcome of the case, as per protocol. Subsequently, Norton's part-time teaching contract was not renewed for future semesters and he is no longer employed by King's University College.
His next appearance will be March 7. Sentencing will be held at a later time.
David Norton, a part-time history professor at King's University College, is seen in this undated image.
Kincardine man killed in early morning crash near Port Albert


text:
\section{Introduction}\label{sec:introduction}
A graph $G = (V,E)$ is called \emph{hypohamiltonian} if it is not Hamiltonian, but the deletion of any single vertex $v \in V$ gives a Hamiltonian graph. The smallest hypohamiltonian graph is the Petersen graph, and all hypohamiltonian graphs with fewer than 18 vertices have been classified \cite{AMW}: there is one such graph for each of the orders 10, 13, and 15, four of order 16, and none of order~17. Moreover, hypohamiltonian graphs exist for all orders greater than or equal to 18.

Chv\'{a}tal \cite{C} asked in 1973 whether there exist \emph{planar} hypohamiltonian graphs, and there was a conjecture that such graphs might not exist \cite{Gru}. However, an infinite family of planar hypohamiltonian graphs was later found by Thomassen \cite{T4}, the smallest among them having order 105. This result was the starting point for work on finding the smallest possible order of such graphs, which has led to the discovery of planar hypohamiltonian graphs of order 57 (Hatzel \cite{H} in 1979), 48 (C.~Zamfirescu and T.~Zamfirescu \cite{ZZ} in 2007), and 42 (Wiener and Araya \cite{WA} in 2011). These four graphs are depicted in Figure~\ref{fig:records}.

\begin{figure}[htbp]
\begin{center}
\begin{minipage}[b]{0.24\linewidth}
\centering
\includegraphics[height=3.2cm]{105Thomasson}
\end{minipage}
\begin{minipage}[b]{0.24\linewidth}
\centering
\includegraphics[height=3.2cm]{57Hatzel}
\end{minipage}
\begin{minipage}[b]{0.24\linewidth}
\centering
\includegraphics[height=3.2cm]{48ZZ}
\end{minipage}
\begin{minipage}[b]{0.24\linewidth}
\centering
\includegraphics[height=3.2cm]{42WA}
\end{minipage}
\end{center}
\caption{Planar hypohamiltonian graphs of order 105, 57, 48, and 42}
\label{fig:records}
\end{figure}

Grinberg \cite{G} proved a necessary condition for a plane graph to be Hamiltonian. All graphs in Figure~\ref{fig:records} have the property that one face has size 1 modulo 3, while all other faces have size 2 modulo 3. Graphs with this property are natural candidates for being hypohamiltonian, because they do not satisfy Grinberg's condition. However, we will prove that this approach cannot lead to hypohamiltonian graphs of order smaller than 42. Consequently we seek alternative methods for finding planar hypohamiltonian graphs. In particular, we construct a certain subset of graphs with girth 4 and a fixed number of 4-faces in an exhaustive way. This collection of graphs turns out to contain 25 planar hypohamiltonian graphs of order 40.

In addition to finding record-breaking graphs of order 40, we shall prove that planar hypohamiltonian graphs exist for all orders greater than or equal to 42 (it is proved in \cite{WA} that they exist for all orders greater than or equal to~76). Similar results are obtained for \emph{hypotraceable} graphs, which are graphs that do not contain a Hamiltonian path, but the graphs obtained by deleting any single vertex do contain such a path. We show that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to~156; the old records were 162 and 180, respectively~\cite{WA}.

T.~Zamfirescu defined $\overline{C^i_k}$ and $\overline{P^i_k}$ to be the smallest order for which there is a planar $k$-connected graph such that every set of $i$ vertices is disjoint from some longest cycle and path, respectively \cite{Z2}. Some of the best bounds known so far were $\overline{C^1_3}\leq 42$, $\overline{C^2_3}\leq 3701$, $\overline{P^1_3}\leq 164$ and $\overline{P^2_3}\leq 14694$, which were found based on a planar hypohamiltonian graph on 42 vertices \cite{WA}. We improve upon these bounds using our graphs to $\overline{C^1_3}\leq 40$, $\overline{C^2_3}\leq 2625$, $\overline{P^1_3}\leq 156$ and $\overline{P^2_3}\leq 10350$.

The paper is organized as follows. In Section~\ref{sec:grinbergian} we define Grinbergian graphs and prove theorems regarding their hypohamiltonicity. In Section~\ref{sec:generation} we describe generation of certain planar graphs with girth 4 and a fixed number of 4-faces, and show a summary of hypohamiltonian graphs found among them. In Section~\ref{sec:corollaries} we present various corollaries based on the new hypohamiltonian graphs. The paper is concluded in Section~\ref{sec:con}.

\section{Grinbergian graphs}\label{sec:grinbergian}
Consider a plane hypohamiltonian graph $G = (V,E)$, and let $\kappa(G)$, $\delta(G)$, and $\lambda(G)$ denote the vertex-connectivity, minimum degree, and edge-connectivity of $G$, respectively. We will use tacitly the following fact.

\begin{theo}
$\kappa(G) = \lambda(G) = \delta(G) = 3$.
\end{theo}

\begin{proof}
Since the deletion of any vertex in $V$ gives a Hamiltonian graph, we have $\kappa(G) \geq 3$. Tutte \cite{Tu2} proved that every 4-connected planar graph is Hamiltonian, so $\kappa(G) \leq 3$. Thomassen \cite{T3} showed that $V$ must contain a vertex of degree 3, so $\delta(G) \leq 3$. The result now follows from Whitney's Theorem \cite[Theorem~4.1.9]{W}.
\end{proof}

The set of vertices adjacent to a vertex $v$ is denoted by $N(v)$. Let $n = |V|$, $m = |E|$, and $f$ be the number of faces of the plane graph $G$. They satisfy Euler's formula $n-m+f=2$. A $k$-\emph{face} is a face of $G$ bounded by $k$ edges. We define $$I_j := \{ i \ge 3 : i \equiv j \bmod 3 \},$$ and let ${\cal P}_j$ be the family of $k$-faces with $k \in I_j$.

\begin{theo}[{Grinberg's Theorem \cite[Theorem 7.3.5]{W}}]
\label{thm:grinberg}
Given a loopless plane graph with a Hamiltonian cycle\/ $C$ and\/ $f_i$ ($f'_i$)\/ $i$-faces inside (outside) of\/ $C$, we have $$\sum_i (i-2)(f_i - f'_i) = 0.$$
\end{theo}

We call a graph \emph{Grinbergian} if it is 3-connected, planar and of one of the following two types.

\begin{description}
\item[Type 1] Every face but one belongs to ${\cal P}_2$.
\item[Type 2] Every face has even order, and the graph has odd order.
\end{description}

The motivation behind such a definition is that Grinbergian graphs can easily be proven to be non-Hamiltonian using Grinberg's Theorem. Namely, their face sizes are such that the sum in Grinberg's Theorem cannot possibly be zero. Thus, they are good candidates for hypohamiltonian graphs.

Our definition of Grinbergian graphs contains two types. One could ask, if there are other types of graphs that can be guaranteed to be non-Hamiltonian with Grinberg's Theorem based on only their sequence of face sizes. The following theorem shows that our definition is complete in this sense.

\begin{theo}
\label{thm:grinbergian_motivation}
Consider a $3$-connected simple planar graph with\/ $n$ vertices ($n \leq 42$) and\/ $F_i$ \mbox{$i$-faces} for each~$i$. Then there are non-negative integers $f_i,f_i'$ ($f_i+f_i'=F_i$) satisfying the equation\/ $\sum_i (i-2)(f_i - f'_i) = 0$ if and only if the graph is not Grinbergian.
\end{theo}

\begin{proof}
Since the graph is simple and 3-connected, every face must have at least $3$ edges. Applying \cite[Theorem 6.1.23]{W} to the dual of the graph gives $\sum_i F_i = 2e \leq 6f-12$, where $f$ is the number of faces. Thus, the average face size is at most $6-(12/f)$. In addition, the size of a face has to be smaller than or equal to the number of vertices in the graph.

Given a sequence of face sizes $F_i$, the problem of finding coefficients $f_i,f_i'$ that satisfy the equation can be reduced to a simple knapsack problem. Namely, note that $\sum_i (i-2)(f_i - f'_i) = \sum_i (i-2)(F_i - 2f'_i) = \sum_i (i-2)F_i - \sum_i 2(i-2)f'_i$, so solving the equation corresponds to solving an instance of the knapsack problem where we have $F_i$ objects of weight $2(i-2)$, and we must find a subset whose total weight is $\sum_i (i-2)F_i$. The result can be then verified with an exhaustive computer search over all sequences of face sizes that fulfill the above restrictions.
\end{proof}

It should be noted that the result in Theorem \ref{thm:grinbergian_motivation} most likely holds for all $n$, but for our purposes it suffices to prove it for $n \leq 42$.

By Grinberg's Theorem, Grinbergian graphs are non-Hamiltonian. Notice the difference between our definition and that of Zaks \cite{Z}, who defines \emph{non-Grinbergian} graphs to be graphs with every face in ${\cal P}_2$. We call the faces of a Grinbergian graph not belonging to ${\cal P}_2$ \emph{exceptional}.

\begin{theo}
\label{theo:grinbergian_hypohamiltonian_type}
Every Grinbergian hypohamiltonian graph is of Type 1, its exceptional face belongs to\/ ${\cal P}_1$, and its order is a multiple of\/ $3$.
\end{theo}

\begin{proof}
Let $G$ be a Grinbergian hypohamiltonian graph. There are two possible cases, one for each type of Grinbergian graphs.

{\bf Type 1:}
Let the $j$-face $F$ be the exceptional face of $G$ (so $j \notin I_2$), and let $v$ be a vertex of $F$. Vertex $v$ belongs to $F$ and to several, say $h$, faces in ${\cal P}_2$. The face of $G - v$ containing $v$ in its interior has length $3h + j - 2\pmod 3$, while all other faces have length $2\pmod 3$. Since $G$ is hypohamiltonian, $G - v$ must be Hamiltonian. Thus, $G - v$ cannot be a Grinbergian graph, so $3h + j - 2 \in I_2$, whence $j \in I_1$.

{\bf Type 2:}  As $G$ contains only cycles of even length, it is bipartite. A bipartite graph can only be Hamiltonian if both of the parts have equally many vertices. Thus, it is not possible that $G-v$ is Hamiltonian for every vertex $v$, so $G$ cannot be hypohamiltonian and we have a contradiction.

Hence, $G$ is of Type~1, and its exceptional face is in ${\cal P}_1$. Counting the edges we get $2m \equiv 2(f-1)+1 \pmod 3$, which together with Euler's formula gives $$2n = 2m-2f+4 \equiv 2f-1-2f+4 \equiv 0 \!\!\!\pmod 3,$$ so $n$ is a multiple of 3.
\end{proof}

\begin{lem}\label{lem:grinberg_type_1_degree_4}
In a Grinbergian hypohamiltonian graph\/ $G$ of Type 1, all vertices of the exceptional face have degree at least\/ $4$.
\end{lem}

\begin{proof}
Denote the exceptional face by $Q$. Now assume that there is a vertex $v \in V(Q)$ with degree 3, and consider the vertex $w \in N(v) \setminus V(Q)$. (Note that $N(v) \setminus V(Q) \ne \emptyset$, because $G$ is $3$-connected.) Let $k$ be the degree of~$w$. Now consider the graph $G'$ obtained by deleting $w$ from $G$. Denote the number of vertices in the faces of $G$ that contain $w$ by $N_i$ ($1\leq i \leq k$); we have $N_i \equiv 2 \pmod 3$. The number of vertices in the face of $G'$ containing $w$ in its interior is now $m = \sum_i (N_i-2) \equiv 0 \pmod 3$. Assume that $G'$ is Hamiltonian. The graph $G'$ contains only faces in ${\cal P}_2$ except for one face in ${\cal P}_1$ and one in ${\cal P}_0$. The face in ${\cal P}_1$ and the face in ${\cal P}_0$ are on different sides of any Hamiltonian cycle in $G'$, since the cycle must pass through~$v$. The sum in Grinberg's Theorem, modulo 3, is then $(m - 2) + 1 \equiv 2 \pmod 3$ or $-(m - 2) - 1 \equiv 1 \pmod 3$, so $G'$ is non-Hamiltonian and we have a contradiction.
\end{proof}

In Section \ref{sec:generation}, we will use these properties to show that the smallest Grinbergian hypohamiltonian graph has 42 vertices.

\section{Generation of 4-face deflatable hypohamiltonian graphs}\label{sec:generation}
We define the operation \emph{4-face deflater} denoted by $\mathcal{FD}_4$ which squeezes a 4-face of a plane graph into a path of length 2 (see Figure~\ref{fig:operation}). The inverse of this operation is called \emph{2-path inflater} which expands a path of length 2 into a 4-face and is denoted by $\mathcal{PI}_2$. In Figure~\ref{fig:operation} each half line connected to a vertex means that there is an edge incident to the vertex at that position and a small triangle allows zero or more incident edges at that position. For example $v_3$ has degree at least 3 and 4 in Figures~\ref{fig:operation_result} and \ref{fig:operation_original}, respectively. The set of all graphs obtained by applying $\mathcal{PI}_2$ and $\mathcal{FD}_4$ on a graph $G$ is denoted by $\mathcal{PI}_2(G)$ and $\mathcal{FD}_4(G)$, respectively.

\begin{figure}[htbp]
\begin{center}
\begin{minipage}[b]{0.3\linewidth}
\centering
\subfloat[]{\label{fig:operation_result}
\resizebox{0.9\width}{0.9\height}{
	\begin{picture}(0,0)%
	\includegraphics{operation_result.pdf}%
	\end{picture}%
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	\put(-972,-1225){\makebox(0,0)[lb]{\smash{{\SetFigFont{8}{9.6}{\rmdefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$v_5$}%
	}}}}
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}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\resizebox{0.85\width}{0.85\height}{
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	\includegraphics{operation_label.pdf}%
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\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\end{minipage}
\subfloat[]{\label{fig:operation_original}
\resizebox{0.9\width}{0.9\height}{
	\begin{picture}(0,0)%
	\includegraphics{operation_original.pdf}%
	\end{picture}%
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	}}}}
	\end{picture}%
}
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\end{center}
\caption{Operations $\mathcal{FD}_4$ and $\mathcal{PI}_2$}
\label{fig:operation}
\end{figure}

Let $\mathcal{D}_5(f)$ be the set of all simple connected plane graphs with $f$ faces and minimum degree at least 5, which can be generated using the program \emph{plantri} \cite{BM07}. Let us denote the dual of a plane graph $G$ by $G^*$. We define the family of \emph{4-face deflatable graphs} (not necessarily simple) with $f$ 4-faces and $n$ vertices, denoted by $\mathcal{M}^4_f(n)$, recursively as:
\begin{equation}
\mathcal{M}^4_f(n)=\left\{
\begin{array}{lr}
\left\{G^* : G\in\mathcal{D}_5\left(n\right)\right\},&f=0;\\
\\
\bigcup_{G\in\mathcal{M}^4_{f-1}(n-1)}\mathcal{PI}_2(G),&f>0.
\end{array}
\right.
\end{equation}
It should be noted that applying $\mathcal{PI}_2$ to a graph increases the number of both vertices and 4-faces by one. Then, we can filter $\mathcal{M}^4_f$ for possible hypohamiltonian graphs and we define $\mathcal{H}^4_f$ based on it as:
\begin{equation}
\mathcal{H}^4_f(n)=\{G\in\mathcal{M}^4_f(n) : \text{$G$ is hypohamiltonian}\}.
\end{equation}

The function $\mathcal{H}^4_f(n)$ can be defined for $n\geq 20$ because the minimum face count for a simple planar 5-regular is 20 (icosahedron). Also it is straightforward to check that $f\leq n-20$ because $\mathcal{H}^4_f(n)$ is defined based on $\mathcal{H}^4_{f-1}(n-1)$ for $f>0$.

To test hamiltonicity of graphs, we use depth-first search with the following pruning rule: If there is a vertex that does not belong to the current partial cycle, and has fewer than two neighbours that either do not belong to the current partial cycle or are an endpoint of the partial cycle, the search can be pruned. This approach can be implemented efficiently with careful bookkeeping of the number of neighbours that do not belong to the current partial cycle for each vertex. It turns out to be reasonably fast for small planar graphs.

Finally, we define the set of \emph{4-face deflatable hypohamiltonian graphs} denoted by $\mathcal{H}^4(n)$ as:
\begin{equation}
\mathcal{H}^4(n)=\bigcup_{f=0}^{n-20} \mathcal{H}^4_f(n).
\end{equation}

Using this definition for $\mathcal{H}^4_f(n)$, we are able to find many hypohamiltonian graphs which were not discovered so far. The graphs found on 105 vertices by Thomassen \cite{T4}, 57 by Hatzel \cite{H}, 48 by C.~Zamfirescu and T.~Zamfirescu \cite{ZZ}, and 42 by Wiener and Araya \cite{WA} are all 4-face deflatable and belong to $\mathcal{H}^4_0(105)$, $\mathcal{H}^4_1(57)$, $\mathcal{H}^4_1(48)$ and $\mathcal{H}^4_1(42)$, respectively.

We have generated $\mathcal{H}^4_f(n)$ exhaustively for $20\leq n\leq 39$ and all possible $f$ but no graph was found, which means that for all $n<40$ we have $\mathcal{H}^4_f(n)=\emptyset$. For $n>39$ we were not able to finish the computation for all $f$ due to the amount of required time. For $n=40,41,42,43$ we finished the computation up to $f=12,12,11,10$, respectively. The only values of $n$ and $f$ for which $\mathcal{H}^4_f(n)$ was non-empty were $\mathcal{H}^4_5(40)$, $\mathcal{H}^4_1(42)$, $\mathcal{H}^4_7(42)$, $\mathcal{H}^4_4(43)$ and $\mathcal{H}^4_5(43)$. More details about these families are provided in Tables~\ref{tbl:H_40}, \ref{tbl:H_42} and \ref{tbl:H_43}. Based on the computations we can obtain the Theorems~\ref{theo:no_hypo_less_than_40}, \ref{theo:25_hypo_for_40}, \ref{theo:179_hypo_for_42} and \ref{theo:497_hypo_for_43}. The complete list of graphs generated is available to download at \cite{J1}.

\begin{table}[htbp]
\centering
\begin{tabular}{|c|l|l|r|}
\hline
4-Face Count&Face Sequence&Degree Sequence&Count\\
\hline
\multirow{5}{*}{5}&\multirow{5}{*}{$5\times 4, 22\times 5$}&$30\times 3, 10\times 4$&4\\
\cline{3-4}
&&$31\times 3, 8\times 4, 1\times 5$&10\\
\cline{3-4}
&&$32\times 3, 6\times 4, 2\times 5$&9\\
\cline{3-4}
&&$33\times 3, 4\times 4, 3\times 5$&2\\
\cline{3-4}
&&All&25\\
\hline
\end{tabular}
\caption{Facts about $\mathcal{H}^4_5(40)$}
\label{tbl:H_40}
\end{table}

\begin{table}[htbp]
\centering
\begin{tabular}{|c|l|l|r|}
\hline
4-Face Count&Face Sequence&Degree Sequence&Count\\
\hline
\multirow{2}{*}{1}&\multirow{2}{*}{$1\times 4, 26\times 5$}&$34\times 3, 8\times 4$&5\\
\cline{3-4}
&&$35\times 3, 6\times 4, 1\times 5$&2\\
\hline
\multirow{9}{*}{7}&\multirow{9}{*}{$7\times 4, 22\times 5$}&$30\times 3, 12\times 4$&4\\
\cline{3-4}
&&$31\times 3, 10\times 4, 1\times 5$&28\\
\cline{3-4}
&&$32\times 3, 8\times 4, 2\times 5$&57\\
\cline{3-4}
&&$33\times 3, 6\times 4, 3\times 5$&49\\
\cline{3-4}
&&$33\times 3, 7\times 4, 1\times 5, \times 6$&11\\
\cline{3-4}
&&$34\times 3, 4\times 4, 4\times 5$&10\\
\cline{3-4}
&&$34\times 3, 5\times 4, 2\times 5, 1\times 6$&5\\
\cline{3-4}
&&$34\times 3, 6\times 4, 2\times 6$&6\\
\cline{3-4}
&&$35\times 3, 4\times 4, 1\times 5, 2\times 6$&2\\
\hline
All&All&All&179\\
\hline
\end{tabular}
\caption{Facts about $\mathcal{H}^4_1(42)$ and $\mathcal{H}^4_7(42)$}
\label{tbl:H_42}
\end{table}

\begin{table}[htbp]
\centering
\begin{tabular}{|c|l|l|r|}
\hline
4-Face Count&Face Sequence&Degree Sequence&Count\\
\hline
\multirow{2}{*}{4}&\multirow{2}{*}{$4\times 4, 23\times 5, 1\times 7$}&$36\times 3, 6\times 4, 1\times 6$&1\\
\cline{3-4}
&&$37\times 3, 4\times 4, 1\times 5, 1\times 6$&1\\
\hline
\multirow{15}{*}{5}&\multirow{5}{*}{$5\times 4, 22\times 5, 1\times 8$}&$34\times 3, 9\times 4$&8\\
\cline{3-4}
&&$35\times 3, 7\times 4, 1\times 5$&20\\
\cline{3-4}
&&$36\times 3, 5\times 4, 2\times 5$&19\\
\cline{3-4}
&&$37\times 3, 3\times 4, 3\times 5$&1\\
\cline{3-4}
&&$37\times 3, 4\times 4, 1\times 5, 1\times 6$&1\\
\cline{2-4}
&\multirow{10}{*}{$5\times 4, 24\times 5$}&$32\times 3, 11\times 4$&52\\
\cline{3-4}
&&$33\times 3, 9\times 4, 1\times 5$&148\\
\cline{3-4}
&&$34\times 3, 7\times 4, 2\times 5$&175\\
\cline{3-4}
&&$34\times 3, 8\times 4, 1\times 6$&2\\
\cline{3-4}
&&$35\times 3, 5\times 4, 3\times 5$&56\\
\cline{3-4}
&&$35\times 3, 6\times 4, 1\times 5, 1\times 6$&6\\
\cline{3-4}
&&$36\times 3, 3\times 4, 4\times 5$&1\\
\cline{3-4}
&&$36\times 3, 4\times 4, 2\times 5, 1\times 6$&4\\
\cline{3-4}
&&$37\times 3, 2\times 4, 3\times 5, 1\times 6$&1\\
\cline{3-4}
&&$37\times 3, 3\times 4, 1\times 5, 2\times 6$&1\\
\hline
All&All&All&497\\
\hline
\end{tabular}
\caption{Facts about $\mathcal{H}^4_4(43)$ and $\mathcal{H}^4_5(43)$}
\label{tbl:H_43}
\end{table}

\begin{theo}\label{theo:no_hypo_less_than_40}
There is no planar\/ $4$-face deflatable hypohamiltonian graph of order less than\/ $40$.
\end{theo}

\begin{theo}\label{theo:25_hypo_for_40}
There are at least\/ $25$ planar\/ $4$-face deflatable hypohamiltonian graphs on\/ $40$ vertices.
\end{theo}

\begin{theo}\label{theo:179_hypo_for_42}
There are at least\/ $179$ planar\/ $4$-face deflatable hypohamiltonian graphs on\/ $42$ vertices.
\end{theo}

\begin{theo}\label{theo:497_hypo_for_43}
There are at least\/ $497$ planar\/ $4$-face deflatable hypohamiltonian graphs on\/ $43$ vertices.
\end{theo}

\begin{lem}\label{lem:fs_simple_dual}
Let\/ $G$ be a hypohamiltonian planar graph whose faces are at least $5$-faces except one which is a\/ $4$-face. Then any\/ $G'$ in\/ $\mathcal{FD}_4(G)$ has a simple dual.

\begin{proof}
As $G$ is a simple 3-connected graph, the dual $G^*$ of $G$ is simple, too. Let $G'\in\mathcal{FD}_4(G)$ and assume to the contrary that $G'^*$ is not simple.

If $G'^*$ has some multiedges, then the fact that $G^*$ is simple shows that either the two faces incident with $v_1v_5$ or with $v_3v_5$ in Figure~\ref{fig:operation_2_cut_result} (we assume the first by symmetry) have a common edge $v_8v_9$ in addition to $v_1v_5$. Let $v_1v_6$ and $v_1v_7$ be the edges adjacent to $v_1v_5$ in the cyclic order of $v_1$. Note that $v_6\neq v_7$ because $d(G';v_1)\geq 3$ by Lemma~\ref{lem:grinberg_type_1_degree_4}. If $v_1$ and $v_8$ were the same vertex, then $v_1$ would be a cut vertex in $G$ considering the closed walk $v_1v_6\cdots v_8(=\!\!v_1)$. But this is impossible as $G$ is 3-connected, so $v_1\neq v_8$. Now we can see that $\{v_1,v_8\}$ is a 2-cut for $G$ considering the closed walk $v_1v_6\cdots v_8\cdots v_7v_1$.

Also, if $G'^*$ has a loop, with the same discussion, we can assume that the two faces incident with $v_1v_5$ are the same but then $v_1$ would be a cut vertex for $G$. Therefore, both having multiedges or having loops violate the fact that $G$ is 3-connected. So the assumption that $G'^*$ is not simple is incorrect, which completes the proof.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[b]{0.39\textwidth}
\centering
\subfloat[$G$]{\label{fig:operation_2_cut_original}
\resizebox{0.85\width}{0.85\height}{
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\end{minipage}
\begin{minipage}[b]{0.20\textwidth}
\centering
\resizebox{0.9\width}{0.9\height}{
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\end{picture}%
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\end{minipage}
\begin{minipage}[b]{0.39\textwidth}
\centering
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\resizebox{0.85\width}{0.85\height}{
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\end{minipage}
\end{center}
\caption{Showing that $\mathcal{FD}_4(G)$ has a simple dual}
\label{fig:2_cut_operation}
\end{figure}
\end{proof}
\end{lem}

\begin{theo}\label{theo:type_1_grnbergian_4_face_deflatable}
Any Type 1 Grinbergian hypohamiltonian graph is\/ $4$-face deflatable. More precisely, any Type 1 Grinbergian hypohamiltonian graph of order\/ $n$ is in\/ $\mathcal{H}^4_0(n)\cup\mathcal{H}^4_1(n)$.

\begin{proof}
Let $G$ be a Type 1 Grinbergian hypohamiltonian graph with $n$ vertices. By Theorem~\ref{theo:grinbergian_hypohamiltonian_type} the exceptional face belongs to $\mathcal{P}_1$ so its size is 4 or it is larger. If the exceptional face is a 4-face, then by Lemma~\ref{lem:grinberg_type_1_degree_4} the 4-face has two non-adjacent 4-valent vertices. So we can apply $\mathcal{FD}_4$ to obtain a graph $G'$ which has no face of size less than 5. So $\delta(G'^*)\geq 5$ and $G'^*$ is a simple plane graph by Lemma~\ref{lem:fs_simple_dual}. Thus $G'^*\in \mathcal{D}_5$ and as a result of the definition of $\mathcal{M}^4_f$, $G'^{**}=G'\in \mathcal{M}^4_0(n-1)$. Furthermore, $G\in \mathcal{M}^4_1(n)$ because $G\in\mathcal{PI}_2(G')$ and as $G$ is hypohamiltonian, $G\in \mathcal{H}^4_1(n)$.

But if the exceptional face is not a 4-face, then by the fact that it is 3-connected and simple, $G^*$ is simple as well and as the minimum face size of $G$ is 5, $\delta(G^*)\geq 5$ which means $G\in\mathcal{M}^4_0(n)$ and so $G\in\mathcal{H}^4_0(n)$.
\end{proof}
\end{theo}

\begin{cor}
The smallest Type 1 Grinbergian hypohamiltonian graph has\/ $42$ vertices and there are exactly\/ $7$ of them on\/ $42$ vertices.

\begin{proof}
By Theorem~\ref{theo:type_1_grnbergian_4_face_deflatable} any Type 1 Grinbergian graph belongs to $\mathcal{H}^4_0(n)\cup\mathcal{H}^4_1(n)$ but according to the results presented in the paragraph preceding Theorem~\ref{theo:no_hypo_less_than_40}, we have $\mathcal{H}^4_0(n)\cup\mathcal{H}^4_1(n)=\emptyset$ for all $n < 42$. So there is no such graph of order less than 42. On the other hand, we have $\mathcal{H}^4_0(42)=\emptyset$ and $|\mathcal{H}^4_1(42)|=7$ which completes the proof.
\end{proof}
\end{cor}

\section{Results}\label{sec:corollaries}
We present one of the planar hypohamiltonian graphs of order 40, discovered by us in Figure~\ref{fig:40-01}, and the other 24 in Figure~\ref{fig:40-all}.

\begin{figure}[htbp]
\begin{center}
\includegraphics[height=2.5cm]{40-01}\\
\end{center}
\caption{A planar hypohamiltonian graph on 40 vertices}
\label{fig:40-01}
\end{figure}

\begin{theo}\label{theo:hypo_40}
The graph shown in Figure~\ref{fig:40-01} is hypohamiltonian.
\end{theo}

\begin{proof}
We first show that the graph is non-Hamiltonian. Assume to the contrary that the graph contains a Hamiltonian cycle, which must then satisfy Grinberg's Theorem. The graph in Figure~\ref{fig:40-01} contains five 4-faces and 22 5-faces. Then $$\sum_i (i-2)(f_i - f'_i) \equiv f'_4-f_4 \equiv 0 \!\!\!\pmod 3,$$ where $f_4+f'_4 = 5$. So $f'_4=1$ and $f_4=4$, or $f'_4=4$ and $f_4=1$. Let $Q$ be the 4-face on a different side from the four others.

Notice that an edge belongs to a Hamiltonian cycle if and only if the two faces it belongs to are on different sides of the cycle. Since the outer face of the embedding in Figure~\ref{fig:40-01} has edges in common with all other 4-faces and its edges cannot all be in a Hamiltonian cycle, that face cannot be $Q$.

If $Q$ is any of the other 4-faces, then the only edge of the outer face in the embedding in Figure~\ref{fig:40-01} that belongs to a Hamiltonian cycle is the edge belonging to $Q$ and the outer face. The two vertices of the outer face that are not endpoints of that edge have degrees 3 and 4, and we arrive at a contradiction as we know that two of the edges incident to the vertex with degree 3 are not part of the Hamiltonian cycle. Thus, the graph is non-Hamiltonian. Finally, for each vertex of the graph, Figure~\ref{fig:40-01-cycles} shows a cycle omitting the vertex.
\end{proof}

\newcommand{16mm}{16mm}
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=16mm]{40-01-00} \includegraphics[height=16mm]{40-01-01} \includegraphics[height=16mm]{40-01-02} \includegraphics[height=16mm]{40-01-03} \includegraphics[height=16mm]{40-01-04} \includegraphics[height=16mm]{40-01-05}
\includegraphics[height=16mm]{40-01-06}
\includegraphics[height=16mm]{40-01-07}
\\\vspace{1mm}
\includegraphics[height=16mm]{40-01-08} \includegraphics[height=16mm]{40-01-09} \includegraphics[height=16mm]{40-01-10} \includegraphics[height=16mm]{40-01-11}
\includegraphics[height=16mm]{40-01-12} \includegraphics[height=16mm]{40-01-13} \includegraphics[height=16mm]{40-01-14} \includegraphics[height=16mm]{40-01-15}
\\\vspace{1mm}
\includegraphics[height=16mm]{40-01-16} \includegraphics[height=16mm]{40-01-17}
\includegraphics[height=16mm]{40-01-18} \includegraphics[height=16mm]{40-01-19} \includegraphics[height=16mm]{40-01-20} \includegraphics[height=16mm]{40-01-21} \includegraphics[height=16mm]{40-01-22} \includegraphics[height=16mm]{40-01-23}
\\\vspace{1mm}
\includegraphics[height=16mm]{40-01-24} \includegraphics[height=16mm]{40-01-25} \includegraphics[height=16mm]{40-01-26} \includegraphics[height=16mm]{40-01-27} \includegraphics[height=16mm]{40-01-28} \includegraphics[height=16mm]{40-01-29}
\includegraphics[height=16mm]{40-01-30} \includegraphics[height=16mm]{40-01-31}
\\\vspace{1mm}
\includegraphics[height=16mm]{40-01-32} \includegraphics[height=16mm]{40-01-33} \includegraphics[height=16mm]{40-01-34} \includegraphics[height=16mm]{40-01-35}
\includegraphics[height=16mm]{40-01-36} \includegraphics[height=16mm]{40-01-37} \includegraphics[height=16mm]{40-01-38} \includegraphics[height=16mm]{40-01-39}
\end{center}
\caption{Vertex-omitting cycles}
\label{fig:40-01-cycles}
\end{figure}

We now employ an operation introduced by Thomassen \cite{T5} for producing infinite sequences of hypohamiltonian graphs. Let $G$ be a graph containing a $4$-cycle $v_1 v_2 v_3 v_4 = C$. We denote by ${\rm Th}(G^C)$ the graph obtained from $G$ by deleting the edges $v_1 v_2$, $v_3 v_4$ and adding a new 4-cycle $v'_1 v'_2 v'_3 v'_4$ and the edges $v_i v'_i$, $1 \le i \le 4$. Wiener and Araya \cite{WA} note that a result in \cite{T5} generalizes as follows, with the same proof.

\begin{lem}\label{lem:TH_G_C_cubic_hypohamiltonian}
Let\/ $G$ be a planar hypohamiltonian graph containing a\/ $4$-face bounded by a cycle\/ $v_1 v_2 v_3 v_4 = C$ with cubic vertices. Then\/ ${\rm Th}(G^C)$ is also a planar hypohamiltonian graph.
\end{lem}

Wiener and Araya use this operation to show that planar hypohamiltonian graphs exist for every order greater than or equal to $76$. That result is improved further in the next theorem.

\begin{theo}\label{theo:3_hypohamiltonian_40_42}
There exist planar hypohamiltonian graphs of order\/ $n$ for every\/ $n \geq 42$.
\end{theo}

\begin{proof}
Figures \ref{fig:40-01}, \ref{fig:records}, and \ref{fig:records2} show plane hypohamiltonian graphs on 40, 42, 43, and 45 vertices, respectively. It can be checked that applying the Thomassen operation to the outer face of these plane graphs gives hypohamiltonian planar graphs with 44, 46, 47, and 49 vertices. By the construction, these graphs will have a 4-face bounded by a cycle with cubic vertices, so the theorem now follows from repeated application of the Thomassen operation and Lemma~\ref{lem:TH_G_C_cubic_hypohamiltonian}.
\end{proof}

\begin{figure}[htbp]
\begin{center}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[height=2.5cm]{43}
\end{minipage}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[height=2.5cm]{45Grinbergian}
\end{minipage}
\end{center}
\caption{Planar hypohamiltonian graphs of order 43 and 45, respectively}
\label{fig:records2}
\end{figure}

Whether there exists a planar hypohamiltonian graph on $41$ vertices remains an open question.

Wiener and Araya \cite{WA} further prove that there exist planar hypotraceable graphs on $162+4k$ vertices for every $k \ge 0$, and on $n$ vertices for every $n \ge 180$. To improve on that result, we make use of the following theorem, which is a slight modification of \cite[Lemma 3.1]{T1}.

\begin{theo}
\label{theo:trace}
If there are four planar hypohamiltonian graphs\/ $G_i = (V_i,E_i)$, $1 \leq i \leq 4$, each of which has a vertex of degree\/ $3$, then there is a planar hypotraceable graph of order\/ $|G_1|+|G_2|+|G_3|+|G_4|-6$.
\end{theo}

\begin{proof}
The result follows from the proof of \cite[Lemma~1]{T1} and the fact that the construction used in that proof (which does not address planarity) can be carried out to obtain a planar graph when all graphs $G_i$ are planar.
\end{proof}

\begin{theo}\label{theo:hypotraceable_154_and_above_155}
There exist planar hypotraceable graphs on\/ $154$ vertices, and on\/ $n$ vertices for every\/ $n \geq 156$.
\end{theo}

\begin{proof}
All the graphs obtained in the proof of Theorem~\ref{theo:3_hypohamiltonian_40_42} have a vertex with degree 3. Consequently, Theorem~\ref{theo:trace} can be applied to those graphs to obtain planar hypotraceable graphs of order $n$ for $n = 40+40+40+40-6 = 154$ and for $n \geq 40+40+40+42-6 = 156$.
\end{proof}

The graphs considered in this work have girth 4. In fact, by the following theorem we know that any planar hypohamiltonian graphs improving on the results of the current work must have girth 3 or 4. Notice that a planar hypohamiltonian graph can have girth at most 5, since a planar hypohamiltonian graph has a simple dual, and the average degree of a simple plane graph is less than 6.

\begin{theo}
There are no hypohamiltonian planar graphs with girth\/ $5$ on fewer than\/ $45$ vertices, and there is exactly one such graph on\/ $45$ vertices.
\end{theo}

\begin{proof}
The program \emph{plantri} \cite{BM07} can be used to construct all planar graphs with a simple dual, girth 5, and up to 45 vertices. By checking these graphs, it turns out that only a single graph of order 45 is hypohamiltonian. That graph, which has an automorphism group of order 4, is shown in Figure~\ref{fig:45-P}.
\end{proof}

\begin{figure}[htbp]
\begin{center}
\includegraphics[height=2.5cm]{45Girth5}\\
\end{center}
\caption{A planar hypohamiltonian graph with girth 5 and 45 vertices}
\label{fig:45-P}
\end{figure}

Let $H$ be a cubic graph and $G$ be a graph containing a cubic vertex $w \in V(G)$. We say that we \emph{insert $G$ into $H$}, if we replace every vertex of $H$ with $G - w$ and connect the endpoints of edges in $H$ to the neighbours of $w$.

\begin{cor}
We have $$\overline{C^1_3}\leq 40, \quad \overline{C^2_3}\leq 2625, \quad \overline{P^1_3}\leq 156 \quad \mbox{ and } \quad \overline{P^2_3}\leq 10350.$$

\begin{proof}
The first of the four inequalities follows immediately from Theorem~\ref{theo:hypo_40}. In the following, let $G$ be the planar hypohamiltonian graph from Fig.~\ref{fig:40-01}.

For the second inequality, insert $G$ into Thomassen's graph $H$ from \cite[p.~38]{T5}. This means that each vertex of $H$ is replaced by $G$ minus some vertex of degree 3. Since every pair of edges in $H$ is missed by a longest cycle \cite{SZ}, in the resulting graph $G'$ any pair of vertices is missed by a longest cycle. This property is not lost if all edges originally belonging to $H$ are contracted.

In order to prove the third inequality, insert $G$ into $K_4$. We obtain a graph in which every vertex is avoided by a path of maximal length.

For the last inequality, consider the graph $H$ from the second paragraph of this proof and insert $H$ into $K_4$, obtaining $H'$. Now insert $G$ into $H'$. Finally, contract all edges which originally belonged to $H'$.
\end{proof}
\end{cor}

\newcommand{16mm}{16mm}
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=16mm]{40-02}
\includegraphics[height=16mm]{40-03}
\includegraphics[height=16mm]{40-04}
\includegraphics[height=16mm]{40-05}
\includegraphics[height=16mm]{40-06}
\includegraphics[height=16mm]{40-07}
\includegraphics[height=16mm]{40-08}
\includegraphics[height=16mm]{40-09}
\\\vspace{1mm}
\includegraphics[height=16mm]{40-10}
\includegraphics[height=16mm]{40-11}
\includegraphics[height=16mm]{40-12}
\includegraphics[height=16mm]{40-13}
\includegraphics[height=16mm]{40-14}
\includegraphics[height=16mm]{40-15}
\includegraphics[height=16mm]{40-16}
\includegraphics[height=16mm]{40-17}
\\\vspace{1mm}
\includegraphics[height=16mm]{40-18}
\includegraphics[height=16mm]{40-19}
\includegraphics[height=16mm]{40-20}
\includegraphics[height=16mm]{40-21}
\includegraphics[height=16mm]{40-22}
\includegraphics[height=16mm]{40-23}
\includegraphics[height=16mm]{40-24}
\includegraphics[height=16mm]{40-25}
\end{center}
\caption{The rest of hypohamiltonian graphs on 40 vertices}
\label{fig:40-all}
\end{figure}

\section{Conclusions}\label{sec:con}
Despite the new planar hypohamiltonian graphs discovered in the current work, there is still a wide gap between the order of the smallest known graphs and the best lower bound known for the order of the smallest such graphs, which is 18 \cite{AMW}. One explanation for this gap is the fact that no extensive computer search has been carried out to increase the lower bound.

It is encouraging though that the order of the smallest known planar hypohamiltonian graph continues to decrease. It is very difficult to conjecture anything about the smallest possible order, and possible extremality of the graphs discovered here. It would be somewhat surprising though if no extremal graphs would have nontrivial automorphisms (indeed, the graphs of order 40 discovered in the current work have no nontrivial automorphisms). An exhaustive study of graphs with prescribed automorphisms might lead to the discovery of new, smaller graphs.

The smallest known \emph{cubic} planar hypohamiltonian graph has 70 vertices~\cite{AW}.  We can hope that the current work inspires further progress in that problem too.

\section*{Acknowledgements}
The first two authors were supported by the Australian Research Council. The work of the third author was supported in part by the Academy of Finland under the Grant No.\ 132122; the work of the fourth author was supported by the same grant, by the GETA Graduate School, and by the Nokia Foundation.

\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv