High seroprevalence of Helicobacter pylori in residents of a hospital for people with severe learning difficulties.
Eradication of Helicobacter pylori with lansoprazole and clarithromycin.
Prevalence of parietal cells in the duodenal bulb & relationship to H pylori infection.
Eradication of Helicobacter pylori lowers acid output in patients with duodenal ulcer.
The relationship between duodenal gastric metaplasia, acid output and H pylori.
AW Harris, DI Pryce, SM Gabe, et al.
Lansoprazole, clarithromycin & metronidazole for one week cures H pylori infection.
Recurrence of duodenal ulcer after H pylori eradication is related to high acid output.
JJ Misiewicz, AW Harris, KD Bardham, et al.
One week triple therapy for Helicobacter pylori: a multi-centre comparative study.
AW Harris, JJ Misiewicz, KD Bardham et al.
Incidence of duodenal ulcer healing after one week of PPI triple therapy for eradication of Helicobacter pylori.
Systematic review and economic evaluation of Helicobacter pylori eradication treatment for non-ulcer dyspepsia.
An assessment of open access referral for PEG in a district general hospital.
Systematic review: antacids H2-receptor antagonists, prokinetics, bismuth and sucralfate therapy for non-ulcer dyspepsia.
Methotrexate for Crohn's disease: experience in a district general hospital.
Prospective audit of colonoscopy quality in Kent and Medway, UK.
EA Russo, AW Harris, S Campbell, et al.
Experience of maintenance infliximab therapy for refractory ulcerative colitis in England.
Survey regarding the use of adalimumab as maintenance therapy in Crohn's disease in England and Ireland.
Revalidation of Gastroenterologists, with or with out sedation!
Parenteral methotrexate for patients with Crohn's Disease: how to develop a high-quality and safe self-administration service.
A prospective study of infliximab withdrawal after 12 months of treatment in patients with Crohn's disease: how will NICE guidance affect patient care?
Ulcerative colitis: Management in adults, children, and young people – concise guidance.
Tacrolimus therapy in moderate to subacute ulcerative proctocolitis: a large single-centre cohort study.
Hitting H. pylori for four.
Helicobacter pylori testing at home.
Treating Helicobacter pylori - the best is yet to come?
Helicobacter pylori and urea breath testing. Can the breath test result predict the outcome of treatment?
Current regimens for treatment of Helicobacter pylori.
Drugs, Diseases and Disorders: Pentasa in ulcerative colitis.
Inflammatory bowel disease: safety of current therapies.
6-mercaptopurine and mesalasine for prevention of relapse after conservative surgery for Crohn's disease.
Certolizumab Pegol for treatment of Crohn's disease.
Science Press Limited, London, 1995 (Second edition 1997).
Blackwell Scientific Publications Limited, Oxford, 1996.
Lecture notes on Helicobacter pylori.
Rapid Science Press Limited, London, 1996.
ABC of Upper Gastrointestinal Tract.
Starting Out in Private Practice.
Development of human anti-murine antibodies in patients following treatment with infliximab.
Parenteral methotrexate in refractory Crohn's disease: experience in a district general hospital.
Waiting list validation for screening and surveillance colonoscopy.
Guided primary care management of inflammatory bowel disease.
Prospective audit of colonoscopy quality in Kent & Medway.
Is conscious sedation practiced safely in patients undergoing colonoscopy.
Audit of 30 day mortality following PEG insertion in a District General Hospital compared with NCEPOD results.
Two year prospective audit of use of liquid diets as "bridge" therapy for active Crohn's disease.
Infliximab as a treatment for ulcerative colitis: 3 years experience in a district general hospital.
Cardiac troponins in patients treated with biological therapy for inflammatory bowel disease.
Retrospective study of outcomes of patients with Crohn's disease treated with early immunosuppression.
Parenteral methotrexate for Crohn's disease: 9 years experience in a district general hospital.
Myenteric plexitis does not predict endoscopic Crohn's disease recurrence 12 months following ileo-colonic resection in district general hospital practice.
Experience on the use of adalimumab as maintenance therapy in Crohn's disease.
A prospective audit of the use of calcineurin inhibitors in patients with refractory ulcerative procto-colitis in a district general hospital.
A prospective audit of the use of tacrolimus in patients with refractory subacute ulcerative colitis in a district general hospital.
Optimising Crohn's disease treatment based on the Rutgeerts' score 12 months after ileo-colic resection.
Five year prospective study of the use of tacrolimus in patients with refractory subacute ulcerative colitis.


text:
Home Travel An Offbeat Snowbird Destination: Canada, eh?
An Offbeat Snowbird Destination: Canada, eh?
Larry MacDonald
Photos by Larry McDonald
If RVers who pack up their motorhomes in the fall and head south in winter are known as "snowbirds," are those who head north called "sunbirds?"
My wife, Sandy, and I have been snowbirding from Canada for the past 10 years to resorts in the U.S. Sun Belt – in our case, at locations in Arizona and California. Last year, we decided to spend the winter in Victoria, with the intention of answering the question: Is it possible to snowbird in Canada?
Two areas within Canada are popular snowbird destinations: Osoyoos in the interior of British Columbia, and Victoria on the southern tip of Vancouver Island. We chose Victoria, British Columbia's capital, because of its cosmopolitan ambiance and proximity to the ocean. It's also said to have the mildest climate in Canada.
Motorhome at West Bay, British Columbia.
Since Victoria is on an island, you need to take a ferry to get there with your motorhome. From mainland British Columbia, the quickest route is BC Ferries' Tsawwassen-to-Sidney (Swartz Bay), dropping you off a half-hour north of Victoria. From Washington, the Black Ball Ferry Line runs from Port Angeles to downtown Victoria. The ferry offers service year-round (except for two weeks in January) and has a 14-foot vertical clearance for motorhomes. Sailing time is 90 minutes. Washington State Ferries run from Anacortes to Sidney once daily in the fall and spring, and twice daily in summer. The trip is approximately three hours long, and there is no service during winter.
Fort Victoria RV Park in spring.
While a dozen or so RV campgrounds are available in the area, we prefer Fort Victoria RV Park, located about 20 minutes from downtown. When we contacted the park in mid-November, their 120 transient, full-service sites were nearly full. Fortunately, a friendly staff person was able to shuffle some later arrivals to accommodate us through April. More than half of our neighbors were from other provinces where below-freezing winter temperatures make RV living unbearable. Collectively, we might be considered a unique breed of snowbirds: those who escape winter's wrath without migrating all the way to the Sun Belt!
Fort Victoria RV Park looks a little different in winter.
Average temperatures in Victoria seldom get below freezing during the winter, and snow accumulation is rare. That said; the winter of 2016-17 was the coldest and snowiest in 20 years. Dampness was our primary concern, evidenced by moisture and mold buildup on our windows and walls. To solve this problem, we purchased a dehumidifier which, together with two small ceramic heaters and our electric fireplace, allowed us to be comfortably dry and warm throughout our stay.
Getting Acquainted
Whenever we arrive in a new city, we take a bus tour to give us the lay of the land. We also stop by the local visitor center to find out about "must do" activities.
A tour bus is a great way to see Victoria.
For our tour, we chose Gray Line's double-decker British-style bus. This 90-minute hop-on hop-off excursion provides an overview of the city's major attractions, beginning at the iconic château-style Fairmont Empress Hotel. From there, we meandered slowly through Chinatown, the oldest in Canada, and stopped briefly at Fisherman's Wharf, Oak Bay Marina, Beacon Hill Park, Craigdarroch Castle and other historic or scenic locations. Running commentary provided a wealth of information about Victoria's early days as a fur trading post and frontier town for gold seekers and coal miners.
Chinatown Victoria, British Columbia.
Victoria has one of the most beautiful harbors in the world, with a well-protected marina and a walkway where buskers entertain to the delight of visitors and locals alike. In addition to our bus tour, we took a 45-minute boat excursion with Victoria Harbour Ferry. Our informative captain spoke about early uses of the harbor, including fishing, logging, shipbuilding, fur trading and even soap making. Once-fertile shorelines populated by First Nationers are now chockfull of plush hotels, colorful float homes, commercial piers and terminal buildings. Passengers disembarking from seaplanes, ferries and cruise ships can enjoy this delightful city by visiting waterfront attractions or by browsing through countless one-of-a-kind shops, boutiques and galleries in the downtown area.
Harbour Ferry in Victoria. Photo courtesy Travel Victoria/Deddeda Stemler
In the 1920s, the city became a popular retirement destination, so much so that Canadian writer Pierre Burton unkindly called Victoria "the place where old people go to visit their parents." No longer! According to the 2016 Canadian census, less than 20 percent of Greater Victoria's population of nearly 370,000 is made up of retired seniors, which increases slightly in winter due to the influx of snowbirds. ("Greater Victoria" includes 13 outlying municipalities, while the city itself has a relatively small population of 85,000.)
Pedder Pay Marina RV Park. Photo courtesy Tourism Victoria
At the downtown visitor center, I learned that during winter, Victoria's three most popular attractions are The Butchart Gardens, the Royal BC Museum and the Parliament Buildings.
American poet Ralph Waldo Emerson said, "The earth laughs in flowers." Nowhere is that laughter more plentiful than at Victoria's century-old Butchart ("Butch-art") Gardens that delights more than a million visitors annually. In early spring, 55 acres of flowering bulbs – including tulips, daffodils and hyacinths – are showcased among colorful shrubs and trees, providing a kaleidoscope of colors. A self-guided tour allowed us to leisurely explore five separate gardens – the Indoor Spring Prelude, Japanese, Italian, Sunken and Mediterranean – each immaculately landscaped with an array of annuals, streams and fountains. Hard to miss is the tall chimney overlooking the Sunken Garden (remnants of a quarry) that provides a stoic reminder of the garden's past as a cement factory.
Butchart Gardens during autumn. Photo courtesy The Butchart Gardens
Snowbirds are doubly fortunate in being able to visit this National Historic Site during the winter holiday season. For five weeks, thousands of colorful lights and dozens of larger-than-life figurines representing "The 12 Days of Christmas" provide an awe-inspiring and magical experience. Our favorite was "The Fourth Day of Christmas," represented by four calling birds … each with a cellphone! Lace up the ice skates and glide across the outdoor rink to the sound of seasonal music, or ride one of 30 hand-crafted animals on the indoor carousel. Finish up with a hot cocoa while listening to carolers.
Royal BC Museum
Established in 1886, the multi-story Royal BC Museum was rated by TripAdvisor as the best of 15 historical and art galleries in Victoria. We spent a full day learning about the natural and human history of Victoria and British Columbia. A First Nations gallery presents artifacts and information about their cultures and ultimate conflicts with European settlers. A separate gallery illustrates how British Columbia was shaped by pioneers, evolving into the economic and political structures of the 21st century.
Another gallery, with expansive life-like dioramas, highlights the impact of nature over the years, from the Ice Age to the abundant forests and changing river deltas of present-day landscapes. The interactive exhibits are fun and educational.
During our visit, we saw a temporary exhibit, "Terry Fox: Running to the Heart of Canada." Fox, who lost part of his right leg to bone cancer at the age of 18, intended to run across Canada to raise money for cancer research. In 1981, 21-year-old Fox dipped his prosthetic leg in the ocean at St. Johns, Newfoundland, commencing his "Marathon of Hope" run. He completed a marathon distance (26 miles) every day for 143 days while inspiring a nation with his courage, integrity and determination. Photographs, maps, videos and artifacts – such as his iconic Ford support van, prosthetic leg, worn out runners and his personal diary – capture a day-in-the-life of his run, his interactions with adoring fans and famous Canadians, and the heart-breaking end at Thunder Bay, Ontario, when cancer spread to his lungs. Although he died within a year, his legacy lives on in annual Terry Fox runs throughout the world, at last count raising more than $700 million in Canada for cancer research. To quote this true Canadian hero, "Dreams are made possible if you try."
Parliament Buildings
Parliament building in Victoria, British Columbia.
Directly across the street from the museum are the majestic parliament buildings, the seat of government in British Columbia since 1898. Guided tours provide visitors with details about the original construction and unique architectural features such as symbolic stained-glass windows, an octagonal copper dome, marble columns and hardwood panels. Our tour guide showed us the regally appointed Legislative Chamber, where elected officials meet to discuss proposed laws, budgets and issues that may impact their constituents. After our tour, we interacted with several video exhibits to learn more about famous Canadian politicians, entertainers and humanitarians.
Victoria has an endless variety of other activities for winter visitors. Whether it's playing tennis, pickleball or curling – or just delighting in the entertainment and cultural scene – there is something here for every RVer. "Monday Magazine" lists daily activities, ranging from folk music and Beethoven concerts to craft shows and beer festivals. For foodies, there is no shortage of restaurants, more per capita than any other Canadian city, from casual pub fare to elegant five-star dining. And as might be expected in a British-inspired city, afternoon tea is oh-so-popular – many restaurants offer a variety of teas combined with gourmet pastries.
Tea and scones at Abkhazi Garden.
To sample this tradition, we visited The Teahouse at Abkhazi (Ab-kazi) Garden and devoured three tiers of delicate finger sandwiches, quiches and pastries, all in an intimate dining room overlooking a beautifully landscaped garden. A choice of exotic teas is served in fine china pots and cups; with advanced notice, their regular decadent menu offers gluten-free and vegetarian options. Victoria, because of its many charming, whimsical and lovingly tended gardens, is deservedly called the "City of Gardens." Afternoon Tea at Abkhazi Garden provides a truly memorable experience for body and soul.
Returning to our question, would we consider staying in Canada a viable option to going south in the winter? We would indeed, at least in Victoria, given the abundance of activities available. And to our surprise and delight, we found the greenery, cool temperatures, and even the rain and snow a refreshing alternative to the barren desert landscapes and the hot, dry conditions that we typically encounter down south.
The author, Larry, and his wife, Sue.
Also, we enjoyed some activities that simply aren't available down south, such as curling, dragon boating and making a snowman! Other nice touches included the presence of typically migratory birds – hummingbirds, robins and geese – and the springtime blossoming of cherry trees scattered throughout the city. On the practical side, our Canadian dollar was only worth about 70 cents U.S. this year, so we no doubt saved money by spending it in Canada. Conversely, one RVer from Montana commented that he was delighted with the purchasing power of the U.S. dollar. As he put it, "More bang for the buck!"
All things considered, snowbirding in Victoria was such a positive experience; we're seriously considering hanging out with the nonmigrating flock again next winter!
Victoria, B.C., RV park in winter.
Larry MacDonaldauthor name
Traveling to Cape Breton Island by RV
Snowbirding in Arizona's Sonoran Desert
Top 10 Snowbirding Tips
Snowbird Resorts
RVs for Snowbirds
Class A's for the Long Haul
Sampling Regina, Saskatchewan
RV Adventures with Explorers and Inventors on Nova Scotia's Cape Breton Island
Edmonton, Canada is a City of Many Faces

redpajama_set_name: RedPajamaCommonCrawl


The Conference will address all aspects of this rapidly burgeoning domain, from genome through organ to whole human and population levels, embracing data driven, mechanistic modelling and simulation, machine learning and combinations thereof. Contributions from academic, clinical and industrial participants alike are welcome.
The conference is organised by the Centre of Excellence (CoE) in Computational Biomedicine (CompBioMed), aimed at nurturing and promoting the uptake and exploitation of high performance computing within the biomedical modelling community (www.compbiomed.eu).
Located at the Institute of Engineering and Technology (IET) London: Savoy Place, the CompBioMed Conference 2019 will combine a venue with innovative catering and spectacular views of the River Thames with an exciting programme of relevant symposia and world-renowned keynote and invited speakers, proposed and secured by our International Organising Committee.
Full information on the conference can be found here.
Alternatively please contact compbiomed2019@ucl.ac.uk for any questions you might have.


Nearly 90% of all jobs require computer and technical literacy, and the traditional education model is failing to deliver the skills needed to compete in the new American job market.
The job-market value of skills and knowledge gained from higher education is not always equal to the cost of that education.
Over the last three decades, the number of minority and female higher education students has doubled. From 2006-2017 the number of enrolling students over the age of 25 is expected to be double that of those under 25.
American high school students are behind their international peers on math and science assessments, and only 16% of all bachelor degrees awarded in the U.S. in 2007 were in math or science.
Technology is making education more accessible through mobile devices, so that students will expect to learn, study, and work whenever and wherever they want to.
Colleges and Universities will become more energy efficient, reduce operating costs through innovative uses of technology and change their curricular offerings to reflect the importance of conservation issues in society.
American workers need to cultivate innovation and leadership skills to remain competitive in the world economy as more than 2000 U.S. companies outsourcing their lower-skilled jobs.
There will be greater teacher/student collaboration without boundaries as non-traditional students bring valuable experiences to their education, and online communication erases the need for classroom and office hours.
Students and faculty will engage more in the Classical concept of invention and exploration of new ideas, which does more than prepare students for existing jobs; it empowers students to create their own careers.
Universities will become increasingly goal oriented, especially in proving the value of the education "product" and its application to the job market. Education will shift from humanities-based to skills-based.


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Which Is Safer: Staying in a Hotel or an Airbnb?
See what experts have to say before you book your next stay.
Katherine-ice-cream-300x225.png
Katherine Plumhoff
which is safer? Hotel or Airbnb
Whether you're planning to travel because you're an essential worker who's heading to a new job, you're struggling with your mental health and need a change of scenery, or you're going outside of town for family, it's important for you to know the safest place to lodge during this pandemic.
While many large hotel chains have announced new cleaning initiatives to help limit their staff and guests from contracting coronavirus (COVID-19), Airbnb has also recently announced in May their new Enhanced Clean program, which involves educational tools and materials that hosts can use in 12 different countries across the world. But even though initiatives have been put in place, is one lodging situation better than another?
To find out if either a hotel or an Airbnb (or other short-term rental options) is safer to stay in, we talked with health experts to see what they had to say. Below is everything you need to keep in mind before making your travel plans.
Understand and accept the risks
First, it's important to recognize that even as some states are opening up and relaxing restrictions, we're still in a pandemic. "The safest thing you can do is to stay home, and that's particularly true for any of us who have health conditions or might live with someone who has health conditions," Dr. Gwen Murphy, Ph.D., MPH, director of epidemiology for at-home testing company LetsGetChecked and former epidemiologist at the U.S. National Cancer Institute, tells HelloGiggles.
But if you want to or need to travel, there are ways to do so safely.
Dr. Murphy suggests approaching your plan as a matter of personal responsibility. Start by understanding the coronavirus rates of where you're coming from and where you're going. Ideally, you'd only travel from a low-risk area to another low-risk area, and you'd make sure that you do everything possible to reduce the risk of you transmitting the virus or becoming infected with it.
Part of that means going and staying in your location for at least two weeks if you can so that you can self-isolate and not run the risk of infecting anyone in the community you're going to. Also, you should be planning for isolation even if you've tested negative, says Dr. Murphy: "Testing may be able to shorten that quarantine period, but because you can be negative one day and positive the next, testing doesn't give you carte blanche to travel."
Know what matters when it comes to transmission
Have you been frantically wiping down every door handle you come into contact with before touching it? That's not a bad idea, but that alone won't save you from getting the virus.
"By far the biggest risk of infection is exposure to other people," says Dr. Murphy. "The chance of you getting infected as a result of touching a contaminated surface is actually pretty small, though it is possible."
Which means that the hotel chains that are advertising their extra-heavy-duty cleaning protocols and disinfecting guarantees aren't addressing the real problem of person-to-person transmission. A perfectly sterile hotel bathroom is a good thing, but if you have to pass through unmasked crowds in the lobby, elevator, and hallway in order to get to it, you haven't avoided the real danger.
The most important thing to consider when choosing your accommodation, according to experts, is how much social distancing it allows for, and there's no one universal answer as to whether hotels or Airbnbs provide more of it.
For example, a hotel that's invested in contactless check-in kiosks and is enforcing social distancing rules in their common spaces—like closing breakfast buffets and not having people from different households share elevators—might be a better option than an Airbnb in an apartment building where residents cluster in the lobby and aren't staying six feet apart. But at the same time, a standalone cabin that you rent on Airbnb and use a lockbox to get into could enable a much greater degree of social distancing than staying in a hotel full of other guests, particularly if you were going to the hotel in order to access high-use common areas like a pool or spa.
"How many other people are going to be sharing that space with you?" is the number one question to ask when it comes to deciding whether a hotel or an Airbnb is a better, safer option, says Dr. Murphy. Choose the option that reduces your contact as much as possible with people who are outside of your household.
While a standalone property with a remote check-in option is probably safer than a heavily-trafficked hotel, that doesn't mean staying in an Airbnb is always the best option.
If you've decided that the best option for you is a hotel, infectious disease physician and researcher Dr. Jeanne Breen suggests asking the staff two more questions before booking your reservation:
• What is their policy on time between check-out of prior guests and check-in of new guests in the same room, and what level of cleaning do they do during that window?
• Are masks required of staff and guests in all common areas such as the lobby, hallways, elevators, and gym?
And upon arrival at your accommodation, whether it's a hotel or an Airbnb, Dr. Breen says it's still a good idea to disinfect high-touch surfaces just in case. "Disinfect the doorknobs, faucet handles, light switches, remotes, counters, lamp switches, appliance handles, and any children's toys or sports gear," she suggests.
Finalize your plans and enjoy your stay
Wherever you stay when you travel, remember to do so safely and take responsibility for your own health and the risk that your presence brings with it.
Dr. Murphy, who lives in Ireland, has traveled locally with her family in the last few months and recommends using ground travel as much as possible. "I would not be keen to take my family on an airplane, but I have taken a car journey with my family and vacationed in a private house," she says. So bring your wipes, wear your mask, and keep clear of anyone that's not part of your inner circle.


TANNER, Ala. – The Indians' bat were red-hot in a 20-2 win in five innings in the first game at Calhoun (Ala.) Community College, but cooled off in timely situations in a 3-3 tie in eight innings after the game was called due to darkness in the nightcap.
The Indians wasted little time jumping on top of Calhoun after scoring five runs in the top of the first inning.
Mattie Meine (Olive Branch) singled and scored on a single by Macey Cox (Troy, Tenn.). Samantha Conley (Mandeville, La.) walked and Summer Cryder (Tupelo) reached on an error before scoring on a single by Kacey Carpenter (Grenada). Cox scored when Maggie Smithey (New Albany) reached on a fielder's choice, who later scored on a single by Jessica Davis (Cumberland).
ICC continued to roll in the top of the second by pushing across eight runs to take a 13-0 lead over the Warhawks. After a pop up to the pitcher to start the inning, the Indians had eight-straight hits in the inning.
Rachel Sternisha (Olive Branch) singled and scored on a double by Conley, who scored on a double by Cryder. Cryder scored on a single by Cox, who scored on a single my Smithey. Carpenter, Smithey, and Hope Harbin (Columbus) scored on a bases-loaded double before the Warhawks could get out of the inning.
Calhoun got a pair of runs back in the bottom of the second to make it a 13-2 contest.
The Indians answered with five runs in the top of the third to make it a 18-2 game.
Cryder walked and scored when Smithey drew a bases-loaded walk. Cox was hit by a pitch and was replaced on the bases by Ellie Menie (Olive Branch), who scored on a sacrifice fly by Harbin. Carpenter was hit by a pitch and scored when Mattie drew a second bases-loaded walk in the inning. Davis walked and scored, along with Mattie, on a single by Emma Grace Schell (Nesbit).
Schell scored on a single by Ellie before Cryder scored on a single by Abigail Colvin (Macon) to make it a 20-2 game in the top of the fifth.
Olivia Burns (Grenada) picked up the win after giving up two runs, both earned, on four hits and walk while striking out four in three innings of work.
Colvin, Ellie, Cox, and Davis each had a pair of hits with Davis driving in four runs in the run-rule victory.
Kaylee Nelson (Moulton, Ala.) and Calhoun's Mckeena Rose put on a pitching clinic the final game of the day.
Nelson gave up three runs, two earned, on three hits and three walks while striking out seven in a no decision. Despite surrendering 12 hits and a pair of earned runs, Rose kept the Indians from coming up with timely hits with runners in scoring position to keep the Warhawks in the game.
ICC left eight runners stranded in scoring position in eight innings of play.
The Indians took a 1-0 lead on a solo home run from Carpenter in the top of the second.
Calhoun came up with a clutch two-out hit from Caroline Parker to score Crystal McElhaney in the bottom of the sixth inning.
Mattie singled to start the seventh inning and advance to second on a sacrifice bunt by Sternisha, but was left stranded after a fly ball to left field and a timely strikeout by Rose ended the inning. The Warhawks put the winning run on second after back-to-back two-out walks, but Nelson picked up her fifth strikeout of the game to get out of the jam.
After a conversation between the coaches and umpires, the game went into extra innings with the international tiebreaker rule.
Cryder started the inning at second base and advanced to third on a sacrifice bunt by Cox before scoring on a double off the wall by Carpenter. Carpenter scored on a single by Smithey, who advanced to second on a single by Joanna Vasquez (Southaven) – both were left stranded after a strikeout and ground ball to first get out of the inning.
Carpenter went 2-for-4 with a run and two RBI.
McElheny started the bottom half of the eighth at second base and advanced to third on a single by Jessica Shaw, who wisely advanced into scoring position on the throw home by the Indians. Nelson picked up back-to-back strikeouts, but Garth delivered a timely double to the gap to push across a pair of runs. The Indians ran down a fly ball to centerfield to end the inning.
The game was called due to darkness with the game ending in a 3-3 tie.
ICC will step back into conference play on Wednesday, March 20 against No. 11 Northeast Mississippi Community College. The First American National Bank opening pitch for game one is scheduled for 2:00 p.m. in Booneville. The games will be available on LetsGoICCTV.com/red.


We have a theory which says it ought to be possible for a tablet and eReader to be combined into a single device that does both jobs really, really well, for an attractive price. Arguablythe Kindle Fire is the closest we'vecome to that so far, but the Fire isn't available in the UK The Amova 7e G2 is available. and it's billed as 'a tablet design ed for reading' which bodes well. The Amova Te G2 only costs E99, so right away you know it isn't goingto be at the cutting edge.
The specifications give more proof of this. There's 4GB of built-in memory, which isn't too bad, and an SD card slot on the bottom edge lets you add more. Yes. we said SD card slot. Not rnicroSD. A 1GHz processor sits inside, which helps the Dual Touch move fairly quickly, and Wi-Fi gets you on to the intemet, though there's no SIM support.
The build screams budget device being made of rather creaky grey plastic. We do like the two side buttons that double up for page turning when reading eBooks and moving through apps in other situations, but this isn't the kind of ta blet to wow you with its sharp looks. It is a bit heavytoo at 370g, though it fits very nicely in one hand at 190 it 130 at 11.2mm. The screen, meanwhile, measures seven inches and delivers 800 x 480 pixels.
That's low resolution indeed for a tablet these days, and the TFT panel looks every inch a budget quality screen. The re's auto rotate support, but we found the sensor to be very unresponsive and, as a result, auto rotation was hit and miss. Indeed, it didn'twant to work at all for us at times. The touchscreen is resistive which rather surprised us -though using resistive technology is a good way of saving costs. We did n't find it a problem to use for simple sweeping or selection, though it does need a fimrer press than more capacitive screens do.
But its multitouch support is suspect. We found that the standard finger-pinch for zooming into webpages and the like was unresponsive and zooming it self was jerky. There's no pinch to zoom support while eBook reading either. There's no camera or GPS, but if what you want is primarily an eReader with some additional functions, you might not mind this. More irritating, perhaps, is that while the Arnova 7e G2 runs Android 2.3, as is often the case with budget tablets it is not the full version.
There are some recognisable Android apps such as email support, Gallery, and the Android music player (the speaker delivers a prettytinny sound by the way). But the Google apps are missing - Google Maps, YouTube, Gmail and, of course, the Market. There's the AppsLib software market instead, which provides a subset ofthe full Google Market.
Arnova also adds a Sudoku app, presumably thinking that's the kind of app an eReader owner might like, and there is more than one reading option on board. You get the Aldiko reader, N00K app and Adobe Reader Mobile. The Kindle app, however, wasn't in the AppsLib store. Several books are provided for you to try out. and on the whole we found the experience reasonable but not wonderful.
The screen is certainly readable and not too harsh on the eye. Butthose bezel buttons don't seem to work in the Aldiko reader, which is irritating, and the whole thing feels more like a flung-together product than the fully rounded eReader and Android tablet mix that we really want to see.


text:
\section{Introduction}
\label{sec:in}
Let $G=(V(G),E(G))$ an undirected graph with vertex set $V(G)$ and edge set $E(G)$. A set $I\subset V$ is called \emph{independent} whenever $I$ has no pair of adjacent vertices.
A \emph{proper $k$-coloring} of $G$ is a function $c\colon V(G)\to \{1,2, \ldots,k\}$ such that each color class, $c^{-1}(i)$, is an independent set for each $i \in \{1,2, \ldots,k\}$. We say that $G$ is \emph{$k$-colorable} if there exists a proper $k$-coloring of $G$.
The \emph{chromatic number} of $G$, denoted by $\chi(G)$, is the minimum $k$ such that $G$ admits a proper $k$-coloring of its vertex set.

Let $D =(V(D),A(D))$ be a loopless directed graph (digraph) with vertex set $V(D)$ and arc set $A(D)$.
A digraph $H$ is a \emph{subdigraph} of $D$ if $V(H)\subset V(D)$ and $A(H)\subset A(D)$ and every arc in $A(H)$ has both end vertices in $V(H)$; if $V(H)=V(D)$ we sill say that $H$ is a \emph{spanning} subdigraph of $D$. The \emph{subdigraph induced} by $S\subset V(D)$, denoted by $D\langle U \rangle$, is the subdigraph of $D$ whose vertex set is $U$ and whose arc set consists of all arcs in $D$ that have both end vertices in $U$.
A digraph $D$ is \emph{strong} if, for every pair $u$, $v$ of distinct vertices in $V(D)$, there exists a directed path from $u$ to $v$. A \emph{strong component} of a digraph $D$ is a maximal induced subdigraph of $D$ which is strong.
A digraph $D$ is said to be \emph{acyclic} if $D$ contains no directed cycle; and a set of vertices $U$ of a digraph $D$ is \emph{acyclic} whenever $D\langle U\rangle$ is an acyclic digraph.
The dichromatic number of a digraph $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits a $k$-coloring of its vertex set in such a way that each color class is acyclic. This number was independently introduced by Mohar and his collaborators in two papers, \cite{Bokal2004227} and \cite{Mohar2003107}, and by \cite{Neumann-Lara1982265}, as a generalization of the chromatic number of graphs, in this way: take a loopless digraph $D$ such that all of its arcs are symmetric and consider the graph $G$ obtained from $D$ by replacing each pair of symmetric arcs by an undirected edge, then the dichromatic number of $D$ is equal to the chromatic number of $G$, as acyclic colorings in $D$ are proper colorings in $G$ and vice versa.

As the dichromatic number of a digraph is a generalization of the chromatic number of an undirected graph, many authors have studied the way to generalize results for proper colorings in graphs to acyclic colorings in digraphs, as \cite{Bokal2004227, Harutyunyan2011170, Harutyunyan20121823, Keevash2013693, Mohar2003107}. In particular, \cite{Chen2007923} proved  that the problem of determining if the dichromatic number of a given digraph $D$ equals 2 is $NP$-complete, even when $D$ is a tournament.

\cite{Bondy1976277} proved that the chromatic number of a digraph $D$ is at most its \emph{circumference}, the length of a longest cycle. Notice that any proper coloring of a loopless digraph $D$ gives adjacent vertices different colors, so it is also an acyclic coloring and thus the chromatic number is an upper bound for the dichromatic number of $D$. In this way, the circumference is also an upper bound for $\chi_A(D)$.

\cite{Neumann-Lara1982265} proved  that
\begin{math}\chi_A(D)=  \max\{ \chi_A(H)\ |\ H \text{ is a strong component of } D\}, \end{math} for any digraph $D$.
Thanks to this, we can focus our attention only on strongly connected digraphs.

Concerning acyclic colorings,  \cite{Neumann-Lara1982265} proved that for any fixed integers $k$ and $r$ with $ k\geq r\geq 2$,
whenever a digraph $D$ contains no cycle of length 0 or 1 modulo $r$, then
\begin{math} \chi_A(D)\leq k. \end{math}
In this direction, \cite{Chen2015210} gave a more general result. They proved for integers $k$ and $r$ with $k\geq 2$ and
\begin{math} k\geq r\geq 1 \end{math}
that: i) if a digraph $D$ contains no cycle of length $1$ modulo $k$, then $D$ can be colored with $k$ colors so that each color class is a stable set; ii) if a digraph $D$ contains no cycle of length $r$ modulo $k$, then $D$ can be colored with $k$ colors so that each color class induces an acyclic subdigraph of $D$; iii) if an undirected graph $G$ contains no cycle of length $r$ modulo $k$, then $G$ is $k$-colorable if $r\neq 2$ and $(k+1)$-colorable otherwise.

\cite{Tuza1992236} used a depth first search (DFS) tree of a graph to study its proper colorings, he proved that if a graph $G$ has an orientation $D$ such that every cycle $C$ of length
\begin{math} l(C)\equiv 1 \pmod{k} \end{math}
contains at least $l(C)/k$ arcs in each direction, then
\begin{math} \chi(G)\leq k; \end{math}
and as a consequence he obtained that if, for some positive integer $k$ ($k\geq 2$), an undirected graph $G$ contains no cycle whose length minus one is a multiple of $k$, then $G$ is $k$-colorable. Moreover, by using the DFS algorithm, if $G$ has $n$ vertices and $m$ edges, then a proper $k$-coloring can be found in $c(n+m)$ steps, for some constant $c$ independent of the value of $k$.
Now we will use the directed version of the DFS algorithm to study the dichromatic number of a digraph.

We will be quoting the following results on the chromatic number of graphs:

\begin{theorem}[\cite{Brooks1941194}]
\label{brooks' theorem}
Let $G$ be a connected graph with maximum degree $\Delta$. If $G$ is neither an odd cycle nor a complete graph, then
\begin{math} \chi(G)\leq \Delta. \end{math}
Otherwise
\begin{math} \chi(G)=\Delta + 1. \end{math}
\end{theorem}

\begin{theorem}[\cite{Tuza1992236}]
\label{tuza}
Let $G$ be a graph; $k$ be an integer such that $k \geq 2$. If there is no cycle of length $1$ modulo $k$ in $G$, then
\begin{math} \chi_A(G)\leq k. \end{math}
\end{theorem}

A digraph $T$ is an \emph{tree} if $T$ is connected, it contains no directed cycle and its underlying graph contains no cycle.
Moreover, $T$ is an \emph{out-tree} if $T$ is a tree with just one vertex $r$ of in-degree zero and any other vertex different from $r$ has in-degree one. The vertex $r$ is the \emph{root} of $T$.
If an out-tree $T$ is a spanning subdigraph of a digraph $D$, $T$ is called an
\emph{out-branching}.

A fixed arbitrary tree $T$ obtained from $D$ by the DFS algorithm will be called a \emph{DFS tree} of $D$, which is an out-branching rooted at a vertex $r$.

In this paper we prove several properties of a DFS tree $T$ of a strongly connected digraph $D$, which allow us to partition $V(D)$ into acyclic sets and in this way we obtain upper bounds for $\chi_A(D)$.
Let $T$ be a DFS tree of a strongly connected digraph $D$ with girth $g$, and let $V_0$, $V_1$, \ldots, $V_t$ be the levels of $T$:
\begin{enumerate}
\item Let $G_T$ be the undirected graph with vertex set
\begin{math} V(G_T)=V(T)=V(D) \end{math}
and $uv\in E(G_T)$ whenever there is a backward arc between $u$ and $v$ in $D$. Then
\begin{math} \chi_A(D)\leq \chi (G_T). \end{math}
\item Let $G^D$ be the undirected graph with vertex set
\begin{math} V(D)=\{V_0, V_1, \ldots, V_t\} \end{math}
and such that $V_iV_j\in E(G^D)$ where $i>j$ if and only if there is a backward arc $(u,v)\in A(D)$ with $u\in V_i$ and $v\in V_j$. Then
\begin{math} \chi_A(D)\leq \chi (G^D). \end{math}
\item Let
\begin{math} k=\lceil{\frac{t+1}{g-1}}\rceil; \end{math}
\begin{math} U_h = \bigcup_{j=0}^{g-2} V_{h(g-1)+j} \end{math}
for each $h\in [k]_0$, where
\begin{math} V_{(k-1)(g-1)+j} = \emptyset \end{math}
whenever
\begin{math} (k-1)(g-1)+j>t; \end{math}
and $G$ be the undirected graph with vertex set
\begin{math} V(G)=\{U_0, \ldots, U_{k-1}\} \end{math}
and $U_iU_j\in E(G)$ whenever $i>j$ and there is a backward arc $(u,v)\in A(D)$ such that $u\in U_i$ and $v\in U_j$. Then
\begin{math} \chi_A(D)\leq \chi (G). \end{math}
\end{enumerate}

We also prove our main results:

\begin{theorem}
\label{theo cycles of length r_i modulo k}
Let $k$ and $s$ be two integers such that $s\geq 1$; let $r_1$, \ldots, $r_s$ be $s$ integers such that
\begin{math} k\geq r_i\geq 0 \end{math}
for each $i\in[s]$ and $D$ a strongly connected digraph such that all cycles in $D$ have length $r$ modulo $k$ for some
\begin{math} r\in\{r_1,\ldots,r_s\}. \end{math}
If $r_i\neq 1$ for each $i\in[s]$, then
\begin{math} \chi_A(D)\leq 2s+1. \end{math}
\end{theorem}

\begin{theorem}
\label{theo 03}
Let $D$ be a strongly connected digraph with girth $g$; $k$ and $p$ be two integers such that
\begin{math} k\geq g-1 \geq p \geq 1. \end{math}
If there is no cycle of length $r$ modulo
\begin{math} \hat{k}=\lceil{\frac{k}{p}}\rceil{p} \end{math}
in $D$ for each
\begin{math} r\in \{-p+2,\ldots, 0,\ldots, p\}, \end{math}
then
\begin{math} \chi_A(D)\leq \lceil{\frac{k}{p}}\rceil. \end{math}
\end{theorem}

\begin{theorem}
\label{circumference and girth}
Let $D$ be a digraph with at least one cycle and let $c$ and  $g$ be its circumference and girth, respectively. Then
\begin{math} \chi_A(D)\leq \lceil{ \frac{c-1}{g-1}}\rceil+1. \end{math}
\end{theorem}

Our results show that if we have more information about the lengths of cycles in a digraph, then we can improve the bounds for the dichromatic number known until now. The restriction on the cycles lengths in Theorems \ref{theo cycles of length r_i modulo k} and \ref{theo 03} provide better bounds. In Theorem \ref{theo cycles of length r_i modulo k} whenever $s<\lfloor \frac{k}{2} \rfloor$, $\chi_A(D)<k$ and in Theorem \ref{theo 03} whenever $k=sp$ with $s\geq 3$, $\chi_A(D)\leq s < k$. Our results are different from those obtained by \cite{Chen2015210}. However, these two cases lower their bound.

Notice that along our proofs we will use the DFS algorithm to obtain acyclic colorings and this algorithm is polynomial. So, we will be able to color digraphs in an ordered fast way.

The text is divided into four sections, including the introduction. In Section \ref{sec:definitions} we will give the definitions and the notation that we will use throughout this document. In section \ref{sec:preliminary results} we will see some properties of a digraph $D$ derived from the structure of a DFS tree of $D$ and we will construct three different graphs from $D$ whose chromatic numbers will bound the dichromatic number of $D$. Finally, in Section \ref{sec:main results} we will prove our three main results: Theorems \ref{theo cycles of length r_i modulo k}, \ref{theo 03} and \ref{circumference and girth}.

\section{Definitions}
\label{sec:definitions}
In this paper $D=(V(D),A(D))$ will denote a loopless digraph. Two vertices $u$ and $v$ are adjacent if $(u,v)$ or $(v,u)$ is in $A(D)$. The \emph{underlying graph} of $D$ is the graph $G$ obtained from $D$ by replacing each arc $(u,v)$ for the undirected edge $uv$ and then deleting all multiple edges between every pair of vertices apart from one.
The subdigraph induced by a set of vertices $U\subseteq V(D)$ will be denoted by
\begin{math} D\langle U\rangle; \end{math}
and if $H$ is a subdigraph of $D$, the subdigraph induced by $V(H)$ will be simply denoted by
\begin{math} D\langle H\rangle. \end{math}

The set of integers $\{1,2, \ldots, k\}$ will be denoted by $[k]$, and  $[k]_0$ will denote the set $\{0$, 1, \ldots, $k-1\}$.

Here the paths and cycles are always directed.
The \emph{girth} (resp. \emph{circumference}) of $D$ is the shortest (resp. longest) cycle.
A \emph{proper $k$-coloring} of $D$ is a proper $k$-coloring of its underlying graph $G$ (this is a coloring of $V(D)$ such that no two adjacent vertices are colored alike). The \emph{chromatic number} of $D$, denoted by $\chi(D)$, is the minimum $k$ such that $D$ admits a proper $k$-coloring.
An \emph{acyclic $k$-coloring} of $D$ is a function
\begin{math} c\colon V(D)\to [k] \end{math}
such that $c^{-1}(i)$ induces an acyclic subdigraph in $D$ for each $i\in [k]$. The \emph{dichromatic number} of $D$, denoted by $\chi_A(D)$, is the minimum $k$ such that $D$ admits an acyclic $k$-coloring.

A subdigraph $T$ of a digraph $D$ is an \emph{out-branching} if: (i) $T$ is connected, it contains no directed cycle and its underlying graph contains no cycle, (ii) $T$ has just one vertex $r$ of in-degree zero and any other vertex different from $r$ has in-degree one, the vertex $r$ is the \emph{root} of $T$ and (iii) $T$ is a spanning subdigraph of $D$ ($V(T)=V(D)$).

A fixed arbitrary tree $T$ obtained from $D$ by the DFS algorithm will be called a \emph{DFS tree} of $D$, which is an out-branching rooted at a vertex $r$.

Recall that: $f(v)$ denotes the time a vertex $v\in V(T)$ is explored by the algorithm and, is called the \emph{DFS label} of $v$; each vertex has a unique DFS label and any two different vertices have different DFS labels;
* Whenever there is a $uv$-path in $T$,
$v$ is a \emph{descendant} of $u$ and $u$ is an \emph{ancestor} of $v$.
Finally, an arc $(u,v)\in A(D)$ is: (i) a  \emph{tree arc} if $(u,v)\in A(T)$, (ii) a  \emph{backward arc} if $u$ is a descendant of $v$, (iii) a
\emph{forward arc} if $u$  is an ancestor of $v$ and (iv) a \emph{cross arc} if $u$ is neither an ancestor nor a descendant of $v$.
For further details we refer the reader to the book of \cite{Bang-Jensen2009}, pages 26-29.

Along this paper, $D$ will be a strongly connected digraph and $T$ will be a DFS tree of $D$ rooted at a vertex $r$ and $V$ will denote $V(D)=V(T)$.

\section{Preliminary results}
\label{sec:preliminary results}

In this section we will see some properties of a digraph $D$ derived from the structure of a DFS tree of $D$ that will be useful in the proofs of the main results; and we will construct three different graphs from $D$ whose chromatic numbers bound the dichromatic number of $D$.

We will start with four properties of a DFS tree of a digraph derived from the definitions. Later we will prove Lemma \ref{lemma backward arc}, that will be the key to construct acyclic sets of vertices in a digraph. After we will see different partitions of the vertex set of a digraph $D$ into acyclic sets that are based on a DFS tree of $D$ and we also see a few (simpler) bounds for the dichromatic number of $D$.

\begin{remark}
\label{DFS 0} It follows from the definitions that in $D$ there are four kinds of arcs: tree arcs, forward arcs, backward arcs and cross arcs. If $(u,v)$ is: (i) a tree arc then $f(u)<f(v)$, (ii) a forward arc then $f(u)<f(v)$, (iii) a backward arc then $f(u)>f(v)$ and (iv) a cross arc then $f(u)>f(v)$ (see Proposition \ref{cross arcs}).

Observe that, as there are only four kinds of arcs, if an arc  $(u,v)\in A(D)$ satisfies $f(u)<f(v)$, then it is either a tree arc or a forward arc; and if $f(u)>f(v)$, then it is either a backward arc or a cross arc.
\end{remark}

\begin{remark}
\label{DFS 1} If $f(u)<f(v)$ and $v$ is an out-neighbor of $u$ in $D$, then there is a $uv$-path in $T$.
Even more, if $f(u)<f(v)$ and $v$ is a descendant of $u$, then for each $w$ such that
\begin{math} f(u)<f(w)<f(v) \end{math}
we have $w$ is also a descendant of $u$. This follows from the definition of the DFS algorithm, as all vertices explored between the time a vertex $u$ is visited for the first time, namely $f(u)$, and before $u$ is completely processed are descendants of $u$ (book of \cite{Gross2006}, page 524).
\end{remark}

\begin{remark}
\label{DFS 2}
Whenever there is a $uv$-path in $T$, this is unique in $T$, $u$ must be an ancestor of $v$ in $T$ and $f(u)<f(v)$; moreover, every walk in $T$ is a path.
\end{remark}

Next proposition was proved in the book of \cite{Gross2006}, page 525, and it stands that each cross arc goes from a branch explored later to a branch explored earlier.

\begin{proposition}
\label{cross arcs}
Let $e$ be a cross arc of a DFS tree from a vertex $u$ to a vertex $v$. Then $f(u)>f(v)$.
\end{proposition}

\begin{lemma}
\label{lemma backward arc}
Each cycle in $D$ contains at least one backward arc.
\end{lemma}
\begin{proof}
Let
\begin{math} C=(u_0,u_1,\ldots,u_{k-1},u_0) \end{math}
be a cycle in $D$.
In what follows the subscripts will be taken modulo $k$.

Consider the set of DFS labels of vertices in $C$, this set consists of $k$ different natural numbers, so we can chose the greatest. Take $M\in [k]_0$ such that $f(u_M) > f(u_j)$ for each $j\neq M$, $j\in [k]_0$. Hence, $f(u_{M-1})< f(u_M)$ and $(u_{M-1},u_M)$ is a tree or a forward arc (see Remark \ref{DFS 0}).

Notice that there is an index $s\in [k]_0$ such that $f(u_{s-1})>f(u_s)$, as $f(u_M)> f(u_{M+1})$. Let $t$ be the first such index starting at $M$ and going on the reverse order of $C$.
Thus, $f(u_{t-1})>f(u_t)$ and
\begin{math} f(u_t)<f(u_{t+1})<\cdots <f(u_{M-1})<f(u_M). \end{math}

Observe that $u_t$ is an ancestor of $u_M$ as $(u_t,u_{t+1})$, \ldots, $(u_{M-1},u_M)$ are arcs in $D$ with $f(u_i)<f(u_{i+1})$ for each $i=t$, $t+1$,\ldots, $M-1$, and thus they are forward or tree arcs, by Remark \ref{DFS 0}.

\begin{claim} $t\neq M$. This is clear as $f(u_{M-1})<f(u_M)$ and $f(u_{t-1})>f(u_t)$ by the definitions of $M$ and $t$.
\end{claim}

\begin{enumerate}[{Case }1:]
\item $t=M+1$. Then $u_t=u_{M+1}$ is an ancestor of $u_M$, and thus $(u_M,u_{M+1})$ is a backward arc (see * in the definitions).
\item $t\neq M+1$ ($t-1\neq M$). Then $u_{t-1}$, $u_t$ and $u_M$ are three different vertices which satisfy
\begin{math} f(u_t)< f(u_{t-1})<f(u_M) \end{math}
and $u_t$ is an ancestor of $u_M$, then by Remark \ref{DFS 1}, $u_{t}$ is an ancestor of $u_{t-1}$ and thus $(u_{t-1},u_{t})\in A(C)$ is a backward arc.
\end{enumerate}
\end{proof}

\begin{definition}
A cycle $C$ in $D$ is an \emph{elementary cycle} whenever there is a backward arc $f=(u,v)\in A(C)$ such that $C-f$ is a $vu$-path contained in $T$.
\end{definition}

A direct consequence of Lemma \ref{lemma backward arc} is that if $c$ is a coloring of a strongly connected digraph $D$, such that end vertices of backward arcs (with respect to some DFS tree) have different colors, then $c$ is an acyclic coloring of $D$. This is clear as all cycles of $D$ contain at least one backward arc, and no backward arc is monochromatic.

We will take advantage of this fact to construct bounds for the dichromatic number of $D$.

\begin{definition}
\label{def:underlying backward graph}
Given  a connected subdigraph $H$ of $T$ define the undirected graph $G_H$ with vertex set $V(G_H)=V(H)$ and $uv\in E(G_H)$ whenever there is a backward arc between $u$ and $v$ in $D$. We will call $G_H$ the \emph{underlying backward graph} of $H$ relative to $T$.
\end{definition}

\begin{lemma}
\label{lemma underlying back graph}
Let $D$ be a strongly connected digraph and $T$ a DFS tree rooted at $r$. Let $H$ be a connected subdigraph of $T$ and  $G_H$ its underlying backward graph relative to $T$. Then
\begin{math} \chi_A(D\langle H \rangle)\leq \chi (G_H). \end{math}
\end{lemma}
\begin{proof}
Let
\begin{math} c\colon V(G_H)\to [\chi(G_H)] \end{math}
be a proper coloring of $V(G_H)$ with $\chi (G_H)$ colors. As $c$ is proper, each color class is a stable set.
This coloring in $V(G_H)=V(H)$ is a coloring of $D\langle H \rangle$ in which the ends of each backward arc have different colors.

By Lemma \ref{lemma backward arc}, any cycle in  $D$ and in particular in $D\langle H \rangle$ contains a backward arc, therefore  $D\langle H \rangle$ has no monochromatic cycle with this coloring. Hence
\begin{math} \chi_A(D\langle H \rangle)\leq \chi (G_H). \end{math}
\end{proof}

\begin{corollary}
Let $D$ be a strongly connected digraph and $T$ a DFS tree rooted at $r$; and $G_T$ its underlying backward graph relative to $T$. Then
\begin{math} \chi_A(D) \leq \chi (G_T). \end{math}
\end{corollary}

\begin{proposition}
\label{proposition chromatic number of a branch}
Let $D$ be a strongly connected digraph and let $T$ be a DFS spanning tree of $D$ with root $r$. Let $x_1$,  \ldots, $x_k$ be the out-neighbors of $r$ in $T$. If $T_1$,  \ldots, $T_k$ are the subtrees of $T$ such that
$x_i\in V(T_i)$,
\begin{math} T=\bigcup_{i=1}^k T_i\end{math}
and
\begin{math} V(T_i) \cap V(T_j)=\{r\} \end{math}
for all $i\neq j$. Then
\begin{math} \chi_A(D)\leq \max_{i\in [k]}\{\chi(G_{T_i})\} \end{math}
for all $i\in [k]$.
\end{proposition}
\begin{proof}
Take
\begin{math} d=\max_{i\in[k]}\{\chi(G_{T_i})\}. \end{math}
Since
\begin{math} \chi(G_{T_i})\leq d \end{math}
for each $i\in [k]$, let
\begin{math} c_i\colon V(G_{T_i})\to [d] \end{math}
be a proper coloring for each $i \in [k]$.
We may assume that $c_i(r)=1$ for each $i\in [k]$, since we can permute two colors to get this situation.

Let's define a color function
\begin{math} c\colon V \to [d+1] \end{math}
on $D$ as follows:
$c(v)=c_i(v)$ for all $v\in V(G_{T_i})$. Notice that
\begin{math} V=\bigcup_{i=1}^k V(G_{T_i})=\bigcup_{i=1}^k V(T_i), \end{math}
\begin{math} V(T_i)\cap V(T_j)=\{r\} \end{math}
whenever $i\neq j$ and $c_i(r)=1$ for each $i\in [k]$, then $c$ is well defined all over $D$.

Let $U_1$, \ldots, $U_d$ be the color classes in $D$. Proceeding by contradiction we will prove that $D\langle U_s\rangle$ is acyclic for all $1\leq s \leq d$.

Suppose that there is a cycle $C$ in $D\langle U_s \rangle$ for some $s\in [k]$. By Lemma \ref{lemma backward arc}, there is a backward arc $(u,v)\in A(C)$ and $c(u)=c(v)$. By definition of backward arc, $u$ is a descendant of $v$ and thus there is an $i\in [k]$ such that
\begin{math} u,v\in V(T_i)=V(G_{T_i}) \end{math}
and $uv\in E(G_{T_i})$. Therefore,
\begin{math} c_i(u)=c(u)=c(v)=c_i(v), \end{math}
which is impossible as $c_i$ is a proper coloring of $G_{T_i}$. Hence,
$D\langle U_s \rangle$ is acyclic for each $s\in [k]$ and $\chi_A(D)\leq d$.
\end{proof}

\begin{notation}
Given a strongly connected digraph $D$ and a DFS tree  $T$ of $D$ rooted at $r$. We will denote by $P_u$  the unique $ru$-path contained in $T$  for each vertex $u\in V$.
\end{notation}

\begin{lemma}
\label{lemma paths between relatives}
Let $D$ be a strongly connected digraph and $T$ be a DFS tree of $D$ rooted at $r$.  If $(u,v)$ is a backward arc, then there is a (unique)  $vu$-path $P$ in $T$ such that
\begin{math} P_u=P_v\cup P \end{math}
and
\begin{math} l(P_u)=l(P_v)+l(P). \end{math}
\end{lemma}
\begin{proof}
Let $(u,v)$ such an arc, so $u$ is a descendant of $v$; by * in the definition of the DFS tree, there is a $vu$-path $P$ in $T$. By Remark \ref{DFS 2}, $P$ is unique.

Now, $P_v\cup P$ is a $ru$-walk which, by Remark \ref{DFS 2}, is a path and
\begin{math} P_u=P_v\cup P \end{math}
with
\begin{math} l(P_u)=l(P_v)+l(P). \end{math}
\end{proof}

\begin{lemma}
\label{lemma partition of V}
Let $D$ be a strongly connected digraph and $T$ be a DFS tree of $D$ rooted at $r$. Let
\begin{math} V_i=\{u\in V\ |\ l(P_u)=i\} \end{math}
for each $i$,
\begin{math} 0\leq i \leq t, \end{math}
where $t$ is the length of a longest path in $T$. The following assertions hold: (i)
\begin{math} V=\bigcup_{i=0}^t V_i, \end{math}
(ii)
\begin{math} V_i\cap V_j=\emptyset \end{math}
for $i\neq j$ and (iii)
\begin{math} D\langle V_i \rangle \end{math}
is acyclic.
\end{lemma}
\begin{proof}
Assertions (i) and (ii) follow directly from the definitions of a DFS tree and the sets $V_i$'s.

In order to prove (iii) assume by contradiction that
\begin{math} D\langle V_i\rangle \end{math}
has a cycle
\begin{math}C=(u_0,u_1,\ldots,u_{k-1},u_0).\end{math}
By Lemma \ref{lemma backward arc}, $C$ has at least one backward arc $(u_, u^+)$ (where $u\in V(C)$ and $u^+$ is its successor in $C$) and, by Lemma \ref{lemma paths between relatives}, there is a $u^+u$-path $P$ in $T$, such that
\begin{math} P_u=P_{u^+}\cup P \end{math}
and
\begin{math} l(P_u)=l(P_{u^+})+l(P)>l(P_{u^+}). \end{math}
Then
\begin{math} i=l(P_u)>l(P_u^+)=i, \end{math}
which is a contradiction.
\end{proof}

\begin{definition}
The set
\begin{math} V_i=\{u\in V\ |\ l(P_u)=i\} \end{math}
will be called the \emph{$i$-th generation} or the \emph{$i$-th level} of $T$, for each $i\in [t+1]_0$, where $t$ is the length of a longest path in $T$; and we will say that $T$ has \emph{length} $t$.
\end{definition}

\begin{remark}
\label{remark backward arc}
If $(u,v)$ is a backward arc, then $u\in V_i$ and $v\in V_j$, with $i>j$. This is a direct consequence of Lemma \ref{lemma paths between relatives}.
\end{remark}

\begin{definition}
\label{def:backward spine}
Let $D$ be a strongly connected digraph $D$ and $T$ a DFS tree of $D$ rooted at $r$ such that $T$ has length $t\geq 1$ and let $V_0$, \ldots, $V_t$ be its generations. Let $G^D$ be the graph with vertex set
\begin{math} V(G^D)=\{V_0,\ldots,V_t\} \end{math}
and such that $V_iV_j \in E(G^D)$ where $i>j$ if and only if there is a backward arc $(u,v)\in A(D)$ with $u\in V_i$ and $v\in V_j$. $G^D$ will be called the \emph{backward spine} of $D$ relative to $T$.
\end{definition}

The chromatic number of $G^D$ is an upper bound for the dichromatic number of $D$ as we will see in the following proposition:

\begin{proposition}
\label{proposition backward spine}
Let $D$ be a strongly connected digraph, $T$ a DFS tree rooted at $r$ and $G^D$ its backward spine relative to $T$. If
\begin{math} c'\colon \{V_0,\ldots,V_t\}\to [\chi(G^D)] \end{math}
is a proper coloring of $G^D$, then the mapping
\begin{math} c\colon V\to [\chi(G^D)], \end{math}
defined as $c(v)=c'(V_i)$ whenever $v\in V_i$, is an acyclic coloring of $D$.
Moreover,
\begin{math}\chi_A(D)\leq \chi(G^D). \end{math}
\end{proposition}
\begin{proof}
Let $d=\chi(G^D)$,
\begin{math} c'\colon V(G^D)\to [d] \end{math}
be a proper coloring of $G^D$ and
\begin{math} c\colon V \to [d] \end{math}
defined as in the hypothesis.

First, $c$ is well defined since $V_0$, \ldots, $V_t$ form a partition of $V(D)$ and thus $c$ assigns a unique color to each vertex $v\in V(D)$.

Second, $c$ is acyclic. Take $C$ an arbitrary cycle in $D$, by Lemma \ref{lemma backward arc}, $C$ contains a backward arc $(u,v)$. Let $i,j\in [t+1]_0$ such that $u\in V_i$ and $v\in V_j$, where $i>j$ by Remark \ref{remark backward arc}. Then, $V_i\neq V_j$, $V_iV_j\in E(G^D)$ and
\begin{math} c'(V_i)\neq c'(V_j). \end{math}
Hence,
\begin{math} c(u)=c'(V_i)\neq c'(V_j)=c(v), \end{math}
and $C$ is not monochromatic in $D$. Since there is no monochromatic cycle in $D$, the color class $c^{-}(i)$ is acyclic for each $i\in [d]$.
Moreover, $c$ is an acyclic coloring of $D$ with $d$ colors, therefore
\begin{math} \chi_A(D)\leq d. \end{math}
\end{proof}

\begin{corollary}
Let $D$ be a strongly connected digraph and $T$ a DFS tree rooted at $r$ such that $T$ has length $t$. Then
\begin{math} \chi_A(D)\leq t+1. \end{math}
\end{corollary}
\begin{proof}
Let $G^D$ the backward spine of $D$ relative to $T$, then
\begin{math} |V(G^D)|=t+1 \end{math}
and
\begin{math} \Delta(G^D)\leq t, \end{math}
where $\Delta(G^D)$ denotes the maximum degree in $G^D$. Therefore, by Theorem \ref{brooks' theorem} and Proposition \ref{proposition backward spine},
\begin{math} \chi_A(D)\leq \chi(G^D)\leq t+1. \end{math}

Moreover, if $G^D$ is neither an odd cycle nor a complete graph, then
\begin{math} \chi_A(D)\leq \chi(G^D)\leq t. \end{math}
\end{proof}

If we have some knowledge about the lengths of cycles in our digraph $D$, we may find a bound for $\chi_A(D)$ by means of Lemma \ref{lemma partition of V}, as we will see.
\cite{Chen2015210} proved for two integers $r$ and $k$ with $k\geq 2$ and  $k\geq r\geq 1$ that if a digraph $D$ contains no cycle of length $r$ modulo $k$, then $\chi_A(D)\leq k$.
We will give other bounds for $\chi_A(D)$ in terms of cycle lengths.

\begin{theorem}[\cite{Chen2015210}]
\label{dichro theo 01}
Let $r$ and $k$ be two integers with $k\geq 2$ and  $k\geq r\geq 1$. If a digraph $D$ contains no cycle of length $r$ modulo $k$, then
$\chi_A(D)\leq k$.
\end{theorem}

\begin{corollary}
\label{corollary dichro theo 01}
Let $D$ be a digraph with girth $g$ and circumference $c$ with
\begin{math} g-1\leq c-g+2 \end{math}
\begin{math} (g\leq \frac{c+3}{2}). \end{math}
Then
\begin{math} \chi_A(D) \leq c-g+2. \end{math}
\end{corollary}
\begin{proof}
Note that $D$ has no cycle of length $g-1$ modulo $c-g+2$. Suppose, by  contradiction, that $C$ is a cycle in $D$ such that
\begin{math} l(C)\equiv g-1 \pmod{c-g+2} \end{math}
then
\begin{math} l(C) = g-1 + q(c-g+2) \end{math}
for some integer $q\geq 0$. If $q=0$ then  $l(C)=g-1$ and this is impossible for a digraph of girth $g$. Therefore, $q\geq 1$ and so
\begin{math} l(C)= g-1 + q(c-g+2)\geq g-1 + c-g+2=c+1, \end{math}
which is a contradiction.

Now, since $D$ has no cycle of length $g-1$ modulo $c-g+2$, by Theorem \ref{dichro theo 01} we have
\begin{math} \chi_A(D)\leq c-g+2. \end{math}
\end{proof}

\begin{remark}
The bound proposed in corollary \ref{corollary dichro theo 01} is sharp.
For any digraph $D$ with at least one cycle
\begin{math} \chi_A(D)\geq 2. \end{math}
Now, take $D$ such that all its cycles have length 3, then
\begin{math} \chi_A(D) \leq 2. \end{math}
Thus, the bound is reached.
\end{remark}

Now we will see a bound for $\chi_A(D)$ when all cycles in $D$ have length $r$ modulo $k$. First, let us prove next Lemma.
which is the directed version of Theorem \ref{tuza} and a particular case of Theorem \ref{dichro theo 01}.

\begin{lemma}
\label{lemma no cycles of length 1 modulo k}
Let $k$ be an integer with $k\geq 2$. Let $D$ be a strongly connected digraph with no cycle of length $1$ modulo $k$; $T$ a DFS tree of length $t$ of $D$ rooted at $r$ and $V_0$, \ldots, $V_t$ its generations. The sets
\begin{math} U_s=\bigcup_{i\equiv s\pmod{k}} V_i \end{math}
for each $s\in [k]_0$ form a partition of $V$ into acyclic sets.
\end{lemma}
\begin{proof}
Take $U_s$ as in the hypothesis for each $s\in [k]_0$.

First,
\begin{math} \bigcup_{s=0}^{k-1} U_s = \bigcup_{i=0}^t V_i=V \end{math}
and
\begin{math} U_i\cap U_j=\emptyset \end{math}
if $i\neq j$ where $i,j\in [k]_0$, by definition.

Second, proceeding by contradiction, assume that $U_s$ is not acyclic for some $s\in[k]_0$, then there is a cycle $C$ in $D$ such that $V(C)\subset U_s$. By Lemma \ref{lemma backward arc}, there is a backward arc $(u,v)\in A(C)$. As $u,v\in U_s$, we have
\begin{math} l(P_u)=i\equiv s \pmod{k} \end{math}
and
\begin{math} l(P_v)=j\equiv s \pmod{k}, \end{math}
by definition of the generations and the $U_s$' and $i>j$, by Remark \ref{remark backward arc}. Hence, by Lemma \ref{lemma paths between relatives}, there is a $vu$-path in $T$, namely $P$, with
\begin{math} l(P)= i-j \equiv 0 \pmod{k} \end{math}
and thus, the elementary cycle, $P\cup(u,v)$, has length 1 modulo $k$, a contradiction.
\end{proof}

In the following Lemma we will see that $g-1$ of consecutive levels in a DFS tree induce an acyclic digraph, where $g$ is the girth of $D$. This will be useful to construct acyclic colorings.

\begin{lemma}
\label{lemma g-1 consecutive generations acyclic}
Let $D$ be a strongly connected digraph with girth $g$; $T$ a DFS tree rooted at $r$ such that $T$ has length $t$ and $V_0$, \ldots, $V_t$ its generations. Then $g-1$ consecutive generations of $T$, $V_h$, $V_{h+1}$, \ldots, $V_{h+g-2}$, induce an acyclic subdigraph in $D$, namely
\begin{math} D\langle \bigcup_{i=0}^{g-2} V_{h+i}\rangle, \end{math}
for each $0\leq h\leq t-g+2$.
Moreover, there is no backward arc with both ends in
\begin{math} \bigcup_{i=0}^{g-2} V_{h+i}. \end{math}
\end{lemma}
\begin{proof}
Let $h\in [t-g+3]_0$.
Suppose by contradiction that there is a cycle $C$ in
\begin{math} D\langle \bigcup_{i=0}^{g-2} V_{h+i} \rangle. \end{math}
By Lemma \ref{lemma backward arc}, $C$ contains a backward arc $(u,v)$. Then $u\in V_{h+i}$ and $v\in V_{h+j}$ for some
\begin{math} 0\leq j < i \leq g-2. \end{math}
Observe that there is a $vu$-path, $P$, in $T$ such that
\begin{math} C'=P\cup(u,v) \end{math}
is an elementary cycle in $D$ and
\begin{math} l(P_u)=l(P_v)+l(P), \end{math}
by Lemma \ref{lemma paths between relatives}. Hence, on the one hand \begin{math} l(P)=l(C')-1 \geq  g-1 \end{math}
and on the other hand
\begin{math} l(P)=l(P_u)-l(P_v)=(h+i)-(h+j)=i-j; \end{math}
therefore, $i-j\geq g-1$, which is a contradiction since
\begin{math} 0\leq j < i \leq g-2 \end{math}
and thus
\begin{math} 0<i-j\leq g-2 < g-1. \end{math}

In this way,
\begin{math} D\langle \bigcup_{i=0}^{g-2} V_{h+i}\rangle \end{math}
is acyclic and there is no backward arc with both ends in
\begin{math}\bigcup_{i=0}^{g-2} V_{h+i}. \end{math}
\end{proof}

\begin{lemma}
\label{lemma partition acyclic sets}
Let $D$ be a strongly connected digraph with girth $g$, $T$ a DFS tree of $D$ rooted at $r$, length $t$ and generations $V_0$, \ldots, $V_t$;
\begin{math} k=\lceil{\frac{t+1}{g-1}}\rceil; \end{math}
and
\begin{math} U_h = \bigcup_{j=0}^{g-2} V_{h(g-1)+j} \end{math}
for each $h\in [k]_0$, where
\begin{math} V_{(k-1)(g-1)+j}=\emptyset \end{math}
whenever
\begin{math} (k-1)(g-1)+j>t. \end{math}
Then $U_0$, \ldots, $U_{k-1}$ form a partition of $V$ into acyclic sets. Moreover, there is no backward arc $(u,v)$ in $D$ with both ends in the same $U_h$.
\end{lemma}
\begin{proof}
First, as $V_0$, \ldots, $V_t$ form a partition of $V$ and
\begin{math} t < \frac{(t+1)}{(g-1)}(g-1) \leq \lceil{\frac{t+1}{g-1}}\rceil(g-1)=k(g-1), \end{math}
it follows that each $i\in [t+1]_0$ satisfies
\begin{math} i\leq t \leq k(g-1)-1=(k-1)(g-1)+g-2 \end{math}
and so, by Euclidean algorithm, there exist unique $h\in[k]_0$ and $j\in[g-1]_0$ such that $i=h(g-1)+j$ and
\begin{math} V_i\subseteq U_h \end{math}
and by the uniqueness of $h$ we have
\begin{math} U_h\cap U_{h'}=\emptyset \end{math}
if $h\neq h'$.
Hence, the sets $U_0$,\ldots, $U_{k-1}$ form a partition of $V$.
Second, from Lemma \ref{lemma g-1 consecutive generations acyclic}, the induced subdigraph
\begin{math} D\langle U_h \rangle = D\langle \bigcup_{j=0}^{g-2} V_{h(g-1)+j} \rangle \end{math}
is acyclic for each $h\in [k]_0$.
The moreover part is consequence of Lemma \ref{lemma g-1 consecutive generations acyclic}.
\end{proof}

\begin{corollary}
Let $D$ be a strongly connected digraph with girth $g$, $T$ a DFS tree rooted and $r$ and $G^D$ the backward spine of $D$. Then the sets
\begin{math} \{ V_{h(g-1)+j}\}_{j=0}^{g-2} \end{math}
are independent in $G^D$ for each  $h\in [k]_0$, where
\begin{math} V_{(k-1)(g-1)+j}=\emptyset \end{math}
whenever
\begin{math} (k-1)(g-1)+j>t. \end{math}
\end{corollary}
\begin{proof}
From Lemma \ref{lemma partition acyclic sets}, we have that the sets
\begin{math} U_h = \bigcup_{j=0}^{g-2} V_{h(g-1)+j} \end{math}
are acyclic for each $h\in [k]_0$, where
\begin{math} V_{(k-1)(g-1)+j} = \emptyset \end{math}
whenever
\begin{math} (k-1)(g-1)+j>t; \end{math}
and by Lemma \ref{lemma partition acyclic sets}, there is no backward arc with both ends in
\begin{math} U_h = \bigcup_{j=0}^{g-2} V_{h(g-1)+j} \end{math}
and thus, by the definition of backward spine, there is no edge with both ends in
\begin{math} \{ V_{h(g-1)+j}\}_{j=0}^{g-2} \end{math}
for each  $h\in [k]_0$.
\end{proof}

Next proposition gives a bound for $\chi_A(D)$, that may no be very good for an arbitrary digraph, but it is better if $D$ is strong and the ratio between the length of a longest path and the girth is not too large.

\begin{proposition}
\label{proposition path/girth}
Let $D$ be a strongly connected digraph. Let $g$ be the girth of $D$ and $l$ be an integer defined as
\begin{math} l=\min_{u\in V}\{j\ |\ j \text{ is the length of a longest path starting at } u\}. \end{math}
Then
\begin{math} \chi_A(D)\leq \lceil{\frac{l+1}{g-1}}\rceil. \end{math}
\end{proposition}
\begin{proof}
Let $r$ be a vertex in $D$ such that  the length of a longest path starting at $r$ in $D$ equals $l$ and let $T$ be a DFS tree of $D$ rooted in $r$, then $T$ has length $t\leq l$. Let $V_0$, \ldots, $V_t$ be the generations of $T$.

Divide the vertex set $V$ in
\begin{math} k = \lceil{\frac{l+1}{g-1}}\rceil \end{math}
subsets as in Lemma \ref{lemma partition acyclic sets}, this is
\begin{math} U_h = \bigcup_{j=0}^{g-2} V_{h(g-1)+j} \end{math}
for each $h\in [k]_0$, where
\begin{math} V_{(k-1)(g-1)+j}=\emptyset \end{math}
whenever
\begin{math} (k-1)(g-1)+j>t. \end{math}

Then
\begin{math} c\colon V\to [k]_0 \end{math}
given by $c(v)=h$ whenever $v\in U_h$ is a $k$-coloring in which each color class is acyclic, by Lemma \ref{lemma partition acyclic sets}. Therefore
\begin{math} \chi_A(D)\leq k. \end{math}
\end{proof}

\begin{corollary}
Let $D$ be a strongly connected digraph of order $n$ and girth $g$.
Then
\begin{math} \chi_A(D)\leq \lceil{\frac{n}{g-1}}\rceil. \end{math}
\end{corollary}
\begin{proof}
The length of a longest path in $D$ is at most $n-1$. Then
\begin{math} \chi_a(D)\leq \lceil{\frac{n}{g-1}}\rceil, \end{math}
by Proposition \ref{proposition path/girth}.
\end{proof}

\begin{definition}
\label{def:condensation graph}
Let $D$, $T$ and $U_0$, \ldots, $U_{k-1}$ be as in Lemma \ref{lemma partition acyclic sets}.
Define the \emph{condensation graph} of $D$ relative to $T$ as the graph, $G$, with vertex set
\begin{math} V(G)=\{U_0,\ldots, U_{k-1}\} \end{math}
and $U_iU_j\in E(G)$ whenever $i>j$ and there is a backward arc $(u,v)\in A(D)$ such that $u\in U_i$ and $v\in U_j$.
\end{definition}

\begin{proposition}
\label{proposition condensed graph}
Let $D$ be a strongly connected digraph, $T$ a DFS tree of $D$ rooted at $r$ and $G$ the condensation graph of $D$ relative to $T$. Then
\begin{math} \chi_A(D)\leq \chi(G). \end{math}
\end{proposition}
\begin{proof}
Let $d=\chi(G)$ and
\begin{math} c\colon V(G)\to [d] \end{math}
a proper coloring of $G$. Define the coloring
\begin{math} c'\colon V\to [d] \end{math}
in $D$ as follows $c'(v)=c(U_h)$ whenever $v\in U_h$.

As $U_0$, \ldots, $U_{k-1}$ is a partition of $V$, $c'$ is well defined and, by Lemma \ref{lemma partition acyclic sets}, each color class is acyclic since each backward arc $(u,v)\in A(D)$ satisfies $u\in U_i$, $v\in U_j$, $i>j$ and $U_iU_j\in E(G)$, hence
\begin{math} c'(u)=c(U_i)\neq c(U_j)=c'(v), \end{math}
as $c$ is proper. Therefore, by Lemma \ref{lemma backward arc}, there is no monochromatic cycle in $D$.
\end{proof}

\section{Main results}
\label{sec:main results}

In this section we will prove our three main results: Theorems \ref{theo cycles of length r_i modulo k}, \ref{theo 03} and \ref{circumference and girth}.

First we prove Theorem \ref{theo cycles of length r_i modulo k}.

\begin{proof}[of Theorem \ref{theo cycles of length r_i modulo k}]
Let $T$ be a DFS tree of $D$ rooted at $\hat{r}$ of length $t$ and $V_0$, \ldots, $V_t$ its generations.

Since $r_i\neq 1$ for each $i\in[s]$ and all cycles in $D$ have length $r$ modulo $k$ for some
\begin{math} r\in\{r_1\ldots,r_s\}, \end{math}
$D$ has no cycle of length 1 modulo $k$, by Lemma \ref{lemma no cycles of length 1 modulo k} the sets
\begin{math} U_s=\bigcup_{i\equiv s\pmod{k}} V_i \end{math}
for each $s\in [k]_0$ form a partition of $V$ into acyclic sets.

Now, consider the graph $H$, with vertex set
\begin{math} V(H)=\{U_1,U_2,\ldots,U_{k-1}\} \end{math}
and $U_iU_j\in E(H)$ if and only if there are $u\in U_i$ and $v\in U_j$ such that either $(u,v)$ or $(v,u)$ is a backward arc in $D$.

Let $U_iU_j\in E(H)$, then there are $u\in U_i$ and $v\in U_j$ such that
\begin{math} l(P_u) \equiv i \pmod{k}, \end{math}
\begin{math} l(P_v) \equiv j \pmod{k} \end{math}
and there is a backward arc between $u$ and $v$.
\begin{enumerate}[{Case }1:]
\item $(u,v)$ is a backward arc in $D$.
Hence, there is a $vu$-path $P$ in $T$, such that $P\cup(u,v)$ is an elementary cycle in $D$. As
\begin{math} l(C)\equiv r_l \pmod{k} \end{math}
for some $l\in[s]$, we have on the one hand
\begin{math} l(P)\equiv r_l-1 \pmod{k} \end{math}
and on the other hand
\begin{math} l(P)=l(P_u)-l(P_v) \equiv i-j \pmod{k}; \end{math}
and thus
\begin{math} i-j\equiv r_l-1 \pmod{k}, \end{math}
or equivalently
\begin{math} i\equiv j+r_l-1 \pmod{k} \end{math}
where
\begin{math} i,j,r_l \in[k]_0. \end{math}
\item $(v,u)$ is a backward arc in $D$.
Hence, there is a $uv$-path $P'$ in $T$, such that $P'\cup(v,u)$ is an elementary cycle in $D$. As
\begin{math} l(C)\equiv r_{l'} \pmod{k} \end{math}
for some $l'\in[s]$, we have on the one hand
\begin{math} l(P')\equiv r_{l'}-1 \pmod{k} \end{math}
and on the other hand
\begin{math} l(P')=l(P_v)-l(P_u) \equiv j-i \pmod{k}; \end{math}
and thus
\begin{math} j-i\equiv r_{l'}-1 \pmod{k}, \end{math}
or equivalently
\begin{math} i\equiv j-(r_{l'}-1) \pmod{k} \end{math}
where
\begin{math} i,j,r_h\in[k]_0. \end{math}
\end{enumerate}
In this way, $U_i$ is adjacent to $U_j$ in $H$ if and only if
\begin{math} i\equiv j+(r_l-1) \pmod{k} \end{math}
or
\begin{math} i\equiv j-(r_l-1) \pmod{k} \end{math}
for some $l\in [s]$.

Since there are no backward arcs with both ends in the same $U_j$ or otherwise $D$ would have a cycle of length 1 modulo $k$, we have $H$ has no loop and the neighborhood of a vertex $U_j$ in $H$ satisfies
\begin{math} N_H(U_j)\subset \bigcup_{l=1}^s\{U_{j-(r_l-1)},U_{j+{r_l-1}}\}, \end{math}
where the subscripts are taken modulo $k$; and thus
\begin{math} d_H(U_j)\leq 2s \end{math}
for each
\begin{math} 0\leq j\leq k-1. \end{math}
Therefore, the maximum degree in $H$ satisfies $\Delta(H)\leq 2s$. By Theorem \ref{brooks' theorem}, $H$ is $(2s+1)$-colorable.

Let
\begin{math} c'\colon V(H) \to [2s+1] \end{math}
a proper coloring in $H$, then
\begin{math} c\colon V \to [2s+1] \end{math}
defined as $c(v)=c'(U_j)$ whenever $v\in U_j$ is an acyclic coloring of $D$.

The mapping $c$ assigns a unique color to each vertex $v\in V$, as the sets $U_1$, \ldots, $U_{k-1}$ form a partition of $V$, so is well defined.

The mapping $c$ is acyclic: let $C$ be an arbitrary cycle in $D$ and $(u,v)\in A(C)$ be a backward arc, then there are different subscripts $i,j \in [k]_0$ such that $u\in U_i$, $v\in U_j$ and thus $U_iU_j \in E(H)$, in this way
\begin{math} c(u)=c'(U_i)\neq c'(U_j)=c(v) \end{math}
as $c'$ is proper. Hence, $C$ is not monochromatic.

Therefore,
\begin{math} \chi_A(D)\leq 2s+1. \end{math}
\end{proof}

If we restrict $s$ in last theorem, then we improve the result in Theorem \ref{dichro theo 01}.

\begin{corollary}
\label{corollary cycles of length r_i modulo k}
Let $k$ and $s$ be two integers such that
\begin{math} s < \lfloor{\frac{k}{2}}\rfloor; \end{math}
let $r_1$, \ldots, $r_s$ be $s$ integers such that
\begin{math} k\geq r_i\geq 0 \end{math}
for each $i\in[s]$ and $D$ a strongly connected digraph such that all cycles in $D$ have length $r$ modulo $k$ for some
\begin{math} r\in\{r_1,\ldots,r_s\}. \end{math}
If $r_i\neq 1$ for each $i\in[s]$, then
\begin{math} \chi_A(D)\leq 2s+1 < k. \end{math}
\end{corollary}

\begin{corollary}
\label{corollary cycles of length r modulo k}
Let $k$ and $r$ be two integers such that
\begin{math} k\geq r\geq 0 \end{math}
and let $D$ be a strongly connected digraph such that every cycle in $D$ has length  $r$ modulo $k$. If $r\neq 1$, then
\begin{math} \chi_A(D)\leq 3. \end{math}
\end{corollary}

Suppose $k$ and $r$ satisfy the hypothesis of corollary \ref{corollary cycles of length r modulo k} and take the worst case scenario in the proof of theorem \ref{theo cycles of length r_i modulo k}, where
\begin{math} N_{H}(U_j)=\{U_{j-(r-1)}, U_{j+r-1}\} \end{math}
for all $0\leq j\leq k-1$, where the subscripts are taken modulo $k$. Then $H$ is a union of
\begin{math} d=\gcd(r-1,k) \end{math}
pairwise disjoint cycles of length
\begin{math} b=\frac{k}{d} \end{math}
of the form
\begin{math} C_j=U_jU_{j+r-1}U_{j+2(r-1)}\cdots U_{j+(b-1)(r-1)}U_j \end{math}
for all $0\leq j< r-1$.**\\
If $b$ is even then, by Theorem \ref{brooks' theorem}, $H$ must be 2-colorable.

** To clarify the statement, consider the bijective mapping
\begin{math} \phi\colon V(H)\to \mathbb{Z}_k \end{math}
given by
\begin{math} U_j\mapsto \bar{j}, \end{math}
where $\bar{n}$ is the number in $[k]_0$ such that
\begin{math} n\equiv \bar{n} \pmod{k}. \end{math}

We know that all the backward arcs with head in $U_j$ have their tail in $U_{j+r-1}$, by Case 1 in the proof of Theorem \ref{theo cycles of length r_i modulo k} and all back arcs having tail in $U_j$ have their head in $U_{j-(r-1)}$, by Case 2 in the proof of Theorem \ref{theo cycles of length r_i modulo k}. Thus, we jump in the opposite direction of the arcs $r-1$ indices, from $U_j$ to $U_{j+r-1}$. If we consider the same jumps in $\mathbb{Z}_k$ we get the automorphism $\psi$ in $\mathbb{Z}_k$ given by
\begin{math} \psi(\bar{n})=\overline{n+(r-1)}. \end{math}
In this way, $U_j$ and $U_l$ are adjacent in $H$ if and only if either
\begin{math} \psi(\bar{j})=\bar{l} \end{math}
or
\begin{math} \psi(\bar{l})=\bar{j}. \end{math}

Notice that, there are $d$ orbits of $\psi$ which are pairwise disjoint, cyclic and have size $t$ and each orbit corresponds to precisely one cycle in $H$. The relationship between $H$ and the automorphism $\psi$ in $\mathbb{Z}_k$ is now clear. \\
Thus, determining a group of the same order as our graph (digraph) and an automorphism on it which represents the adjacencies may help us to color, via its orbits that are always cyclic and mutually disjoint, the vertices of the graph (digraph). However, if the graph (digraph) has more edges (arcs, resp.) than those in the cycles determined by the orbits, this tool may be useless.

\begin{corollary}
Let $k$ and $r$ be two integers such that $k\geq r\geq 0$  and let $D$ be a strongly connected digraph such that every cycle in $D$ has length congruent with $r$ modulo $k$. If $r\neq 1$ and $\frac{k}{d}$ is even, where
\begin{math} d=\gcd(r-1,k), \end{math}
then
\begin{math} \chi_A(D)\leq 2. \end{math}
\end{corollary}
\begin{proof}
This follows from the proof of Theorem \ref{theo cycles of length r_i modulo k} and the explanation in **, noticing that the orbits of $\phi$ as cycles in $H$ have even length
\begin{math} b=\frac{k}{d}. \end{math}
Thus they are 2-colorable and we may color the vertices of $D$ with 2 colors as in the proof of Theorem \ref{theo cycles of length r_i modulo k}.
\end{proof}

Next we will prove Theorem \ref{theo 03}.

\begin{proof}[of Theorem \ref{theo 03}]
Let $T$ be a DFS tree of $D$ rooted at $\hat{r}$ of length $t$ and $V_0$, \ldots, $V_t$ its generations.

Let
\begin{math} k'= \lceil{ \frac{t+1}{p}}\rceil \end{math}
and
\begin{math} U_{h} = \bigcup_{j=0}^{g-2} V_{hp+j} \end{math}
for each $h\in [k']_0$, where
\begin{math} V_{(k'-1)p+j}=\emptyset \end{math}
whenever
\begin{math}(k'-1)p+j>t. \end{math}
In Lemma \ref{lemma g-1 consecutive generations acyclic} we proved each $U_{h}$ is acyclic as it consists of $p\leq g-1$ consecutive generations and that there is no backward arc with both ends in the same $U_{h}$.

These sets form a partition of $V$, since the $V_i$'s form a partition of $V$ and by Euclidean algorithm each $i\in[t+1]_0$ can be written as
\begin{math} i=hp+j \end{math} for unique $h\geq 0$ and
\begin{math} 0\leq j \leq p-1, \end{math}
and thus $V_i \subset U_{h'}$ and
\begin{math} V_i\cap U_{h'}=\emptyset \end{math}
whenever $h'\neq h$.

Define
\begin{math} c\colon V \to [d]_0, \end{math}
where
\begin{math} d = \lceil{\frac{k}{p}}\rceil=\frac{\hat{k}}{p}, \end{math}
as  follows: $c(v)=i$ whenever $v\in U_{h}$ and
\begin{math} h \equiv i \pmod{d}. \end{math}
Then $c$ is an acyclic coloring of $D$.
First, as the sets $U_{h}$ form a partition of $V$, each vertex has exactly one color.
Second, the color classes are acyclic. Suppose by contradiction that there is a monochromatic cycle $C$ in $D$. Then by Lemma \ref{lemma backward arc}, there is a monochromatic backward arc $(u,v)\in A(C)$. Let
\begin{math} h_1, h_2\in [k']_0 \end{math}
and $i\in [d]_0$ such that $u\in U_{h_1}$, $v\in U_{h_2}$ and
\begin{math} h_1\equiv h_2\equiv i \pmod{d}. \end{math}

As there is no backward arc with both ends in the same $U_{h}$, we have $h_1\neq h_2$.
By the definition of the $U_h$'s we have that $u\in V_{h_1 p+j_1}$ for some $j_1\in[p]_0$, $v\in V_{h_2 p+j_2}$ for some $j_2\in[p]_0$,
\begin{math} l(P_u)=h_1p+j_1, \end{math}
\begin{math} l(P_v)=h_2p+j_2 \end{math}
and, as $u$ is a descendant of $v$,
\begin{math} h_1p+j_1>h_2p+j_2, \end{math}
where
\begin{math} h_1,h_2\in[k']_0, \end{math}
\begin{math} j_1,j_2\in[p]_0 \end{math}
and $h_1 > h_2$.

Now, let $P$ be the $vu$-path in $T$. It follows that
\begin{math} l(P)=l(P_u)-l(P_v)=(h_1p+j_1)-(h_2p+j_2)=(h_1-h_2)(g-1)+(j_1-j_2). \end{math}
Since
\begin{math} h_1 \equiv h_2 \pmod{d} \end{math}
and $h_1>h_2$, it follows that
\begin{math} h_1-h_2=qd \end{math}
for some $q\geq 1$ and thus
\begin{math} l(P)= (h_1-h_2)p+(j_1-j_2) = qdp + j_1-j_2 = q\hat{k} + j_1 - j_2 \equiv j_1-j_2 \pmod{\hat{k}}, \end{math}
where
\begin{math} |j_1-j_2|\leq p-1 \end{math}
as
\begin{math} j_1,j_2\in[p]_0. \end{math}

Therefore, the cycle
\begin{math} \gamma=P\cup(u,v) \end{math}
satisfies
\begin{math} l(\gamma)=l(P)+1 \equiv j_1-j_2+1 \pmod{\hat{k}}, \end{math}
where
\begin{math} - (p-1) +1 \leq j_1-j_2+1 \leq (p-1) + 1, \end{math}
or equivalently,
\begin{math} -p+2 \leq j_1-j_2+1 \leq p, \end{math}
which contradicts the hypothesis.

Hence, $c$ is an acyclic coloring and
\begin{math} \chi_A(D)\leq d. \end{math}
\end{proof}

As a consequence of Theorem \ref{theo 03} we have:

\begin{corollary}
Let $D$ be a strongly connected digraph with girth $g$; $k$ and $p$ be two integers such that
\begin{math} k\geq g-1 \geq p \geq 1 \end{math}
and $p \mid k$. If there is no cycle of length $r$ modulo $k$ in $D$ for each
\begin{math} r\in \{-p+2,\ldots, 0,\ldots, p\}, \end{math}
then
\begin{math}\chi_A(D)\leq \frac{k}{p}. \end{math}
\end{corollary}

As the dichromatic number of a digraph with all of its arcs symmetric (girth $g=2$) is equivalent to the chromatic number of an undirected graph, we have as a corollary Tuza's result, Theorem \ref{tuza}.

Now we will prove our last main result, Theorem \ref{circumference and girth}.

\begin{proof}[of Theorem \ref{circumference and girth}]
Let $T$ be a DFS tree of $D$ rooted at $r$ with length $t$ and generations $V_0$, \ldots, $V_t$.  We will partition $V$ into
\begin{math} k=\lceil{ \frac{c-1}{g-1}}\rceil+1 \end{math}
acyclic subsets as follows.
\begin{enumerate}[{Step }1:]
\item Let
\begin{math} k'= \lceil{ \frac{t+1}{g-1}}\rceil \end{math} and
\begin{math} U_{h'} = \bigcup_{j=0}^{g-2} V_{h'(g-1)+j} \end{math}
for each $h'\in [k']_0$, where
\begin{math} V_{(k'-1)(g-1)+j}=\emptyset \end{math}
whenever
\begin{math} (k'-1)(g-1)+j>t. \end{math}
In Lemma \ref{lemma partition acyclic sets} we saw that the sets $U_0$, \ldots, $U_{k'-1}$ form a partition of $V$ into acyclic sets.
\item Give color $h\in[k]_0$ to vertices in $U_{h'}$ whenever
\begin{math} h' \equiv h \pmod{k}, \end{math}
where $h'\in [k']_0$.
The coloring is well defined as the sets $U_0$, \ldots, $U_{k'-1}$ form a partition of $V$ and $h'$ is congruent with exactly one $h\in[k]_0$ modulo $k$; thus, each vertex $v\in V$ is colored with exactly one color.
\end{enumerate}

We will see that each color class is acyclic. Suppose by contradiction that there is a monochromatic cycle $C$. By Lemma \ref{lemma backward arc}, there is a backward arc  $(u,v)\in A(C)$.  As $C$ is monochromatic, there are $h\in [k]_0$ and $i_1,i_2 \in [k']$ such that $u\in U_{i_1}$, $v\in U_{i_2}$ and
\begin{math} i_1 \equiv i_2 \equiv h \pmod{k}. \end{math}

Observe that
\begin{math} u\in V_{i_1(g-1)+j_1} \end{math}
for some $j_1\in[g-1]_0$ and
\begin{math} v\in V_{i_2(g-1)+j_2} \end{math}
for some $j_2\in[g-1]_0$, by the definition of $U_{i_1}$ and $U_{i_2}$. Hence,
\begin{math} l(P_u)=i_1(g-1)+j_1 \end{math}
and
\begin{math} l(P_v)=i_2(g-1)+j_2. \end{math}
Moreover, as $(u,v)$ is a backward arc it follows that
\begin{math} i_1(g-1)+j_1>i_2(g-1)+j_2, \end{math}
where
\begin{math} i_1,i_2\in[k']_0 \end{math}
and
\begin{math} j_1,j_2\in[g-1]_0; \end{math}
and thus $i_1 \geq i_2$. If  $i_1=i_2$, then $u,v\in U_{i_1}$ which is impossible, by Lemma \ref{lemma partition acyclic sets}. Therefore, $i_1>i_2$. Recall
\begin{math} i_1\equiv i_2 \pmod{k}, \end{math} hence
\begin{math} i_1-i_2=qk \end{math}
for some $q\geq 1$.

Now, let $P$ be the $vu$-path in $T$. It follows that
\begin{math} l(P)=l(P_u)-l(P_v)=(i_1(g-1)+j_1)-(i_2(g-1)+j_2)=(i_1-i_2)(g-1)+(j_1-j_2). \end{math}
On the one hand, since
\begin{math} i_1 \equiv i_2 \pmod{k}, \end{math}
we have
\begin{math} l(P)\equiv j_1-j_2 \pmod{k} \end{math}
where
\begin{math} |j_1-j_2|<g-1 \end{math}
as
\begin{math} j_1,j_2\in[g-1]_0; \end{math}
on the other hand, as
\begin{math} i_1-i_2=qk \end{math}
with $q\geq 1$, we have
\begin{math} l(P)= (i_1-i_2)(g-1)+(j_1-j_2) = qk(g-1) +(j_1-j_2)\geq k(g-1) +(j_1-j_2). \end{math}

Consider the cycle
\begin{math} \gamma=P\cup(u,v), \end{math}
we have
\begin{math} g\leq l(\gamma)\leq c \end{math}
and
\begin{math} l(\gamma)=l(P)+1; \end{math}
and thus
\begin{math} g-1\leq l(P)\leq c-1. \end{math}

However,
\begin{math} l(P)\geq k(g-1) +(j_1-j_2)=\left(\lceil{ \frac{c-1}{g-1}}\rceil+1\right)(g-1)+(j_1-j_2) \geq \left( \frac{c-1}{g-1}+1\right)(g-1)+(j_1-j_2) = (c-1+g-1)+(j_1-j_2) > (c+g-2)-(g-1)=c-1, \end{math}
which is impossible. Hence the color classes are acyclic and we have the result.

\end{proof}

The bound of Theorem \ref{circumference and girth} is trivially reached by directed cycles (with no symmetric arcs) and complete digraphs (digraphs where each pair of different vertices induces a directed 2-cycle). And it improves Bondy's result whenever $g\geq 3$.

\acknowledgements
\label{sec:ack}

We would like to thank the anonymous referees for many comments which improved substantially the rewriting of this paper.

\nocite{*}
\bibliographystyle{abbrvnat}

redpajama_set_name: RedPajamaArXiv


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Navin Pokala, Ph.D., assistant professor of Life Sciences at New York Institute of Technology (NYIT) College of Arts and Sciences, with researchers at Caltech, adapted the GAL4-UAS system for expressing transgenes in the nematode C elegans. This system, which uses a gene regulatory protein from yeast, greatly reduces the work required for making cell-specific perturbations. Instead of constructing new DNA and transgenic animals for each cell and actuator, new cell-actuator combinations can be generated by simply mating already-constructed animals, dramatically reducing the time and cost.
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The work is described in Nature Methods was funded by The Howard Hughes Medical Institute and The G. Harold and Leila Y. Mathers Charitable Foundation.
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