text: # Graph Theory - Euler circuit , trail
  
  Consider a complete tripartite graph $K_{\ell,m,n}$
  
  a. Please draw $K_{2,2,3}$.
  b. For general values of $\ell,m$, and $n$, how many vertices are in $K_{\ell,m,n}$?
  c. How many edges are in $K_{\ell,m,n}$?
  d. For what values of $\ell,m$ and $n$ does $K_{\ell,m,n}$ have an Euler circuit?
  e. For what values of $\ell,m$ and $n$ does $K_{\ell,m,n}$ have an Euler trail?
  
  I just know the answer for a , am I correct ? Any ideas for other parts ?
  
  -
  At this link you can start learning how to use $\LaTeX$ to write the mathematics more readably. – Brian M. Scott Dec 13 '12 at 5:29
  
  (a) Your graph is correct, but it would help if you identified the three sets of vertices. They should be grouped as in the diagram below:
  
                             1
  1             1
  
  2                 3
  
  3        2
  
  
  (b) Trivial.
  
  (c) Let the vertex classes of $K_{\ell,m,n}$ be $V_\ell,V_m$, and $V_n$, containing $\ell,m$, and $n$ vertices, respectively. Each vertex in $V_\ell$ is joined by an edge to each vertex in $V_m\cup V_n$, so each vertex in $V_\ell$ has degree $m+n$, and the sum of the degrees of the vertices in $V_\ell$ is $\ell(m+n)$. In similar fashion you can calculate the sums of the degrees of the vertices in $V_m$ and $V_n$ and add them to get the sum of the degrees of all of the vertices in $K_{\ell,m,n}$; then use the handshaking lemma to find the number of edges.
  
  (d, e) A graph has an Euler circuit (or trail) if and only if the degrees of its vertices satisfy a certain condition, and in (c) we saw how to calculate the degrees of the vertices; just combine this information properly.
  
  -
  Thanks,man, really helpful – Node.JS Dec 13 '12 at 7:22
  @Hooman: Glad to help. – Brian M. Scott Dec 13 '12 at 7:25
explanation: The document contains a series of questions related to graph theory, specifically about a complete tripartite graph \( K_{\ell,m,n} \). The questions ask about drawing the graph, determining the number of vertices and edges, and identifying conditions for the existence of Euler circuits and trails. The document also includes a partial answer to the first part of the question and a hint for the other parts.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Graph Theory - Euler circuit , trail
  
  Consider a complete tripartite graph $K_{\ell,m,n}$
  
  a. Please draw $K_{2,2,3}$.
  b. For general values of $\ell,m$, and $n$, how many vertices are in $K_{\ell,m,n}$?
  c. How many edges are in $K_{\ell,m,n}$?
  d. For what values of $\ell,m$ and $n$ does $K_{\ell,m,n}$ have an Euler circuit?
  e. For what values of $\ell,m$ and $n$ does $K_{\ell,m,n}$ have an Euler trail?
  
  I just know the answer for a , am I correct ? Any ideas for other parts ?
  
  -
  At this link you can start learning how to use $\LaTeX$ to write the mathematics more readably. – Brian M. Scott Dec 13 '12 at 5:29
  
  (a) Your graph is correct, but it would help if you identified the three sets of vertices. They should be grouped as in the diagram below:
  
                             1
  1             1
  
  2                 3
  
  3        2
  
  
  (b) Trivial.
  
  (c) Let the vertex classes of $K_{\ell,m,n}$ be $V_\ell,V_m$, and $V_n$, containing $\ell,m$, and $n$ vertices, respectively. Each vertex in $V_\ell$ is joined by an edge to each vertex in $V_m\cup V_n$, so each vertex in $V_\ell$ has degree $m+n$, and the sum of the degrees of the vertices in $V_\ell$ is $\ell(m+n)$. In similar fashion you can calculate the sums of the degrees of the vertices in $V_m$ and $V_n$ and add them to get the sum of the degrees of all of the vertices in $K_{\ell,m,n}$; then use the handshaking lemma to find the number of edges.
  
  (d, e) A graph has an Euler circuit (or trail) if and only if the degrees of its vertices satisfy a certain condition, and in (c) we saw how to calculate the degrees of the vertices; just combine this information properly.
  
  -
  Thanks,man, really helpful – Node.JS Dec 13 '12 at 7:22
  @Hooman: Glad to help. – Brian M. Scott Dec 13 '12 at 7:25
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a series of questions related to graph theory, specifically about a complete tripartite graph \( K_{\ell,m,n} \). The questions ask about drawing the graph, determining the number of vertices and edges, and identifying conditions for the existence of Euler circuits and trails. The document also includes a partial answer to the first part of the question and a hint for the other parts.
  
  Label: multiple-question-and-answer