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Let $m, n$ be nonzero natural numbers. Define $K_{mn}$ to be the complete bipartite graph, which has vertex set $V = V_0 \cup V_1$ such that $ |V_0| = m, |V_1| = n, V_0 \cap V_1 = \emptyset$, and edge set consisting of all edges {$a, b$} with $a \in V_0, b \in V_1$.

a) How many edges does $K_{mn}$ have?
b) What is the average degree of $K_{mn}$?
c) For which values of $m, n$ does $K_{mn}$ have an Euler circuit?

Can someone show me how to solve these? For $K_{mn}$ to be the complete bipartite graph.


marked as duplicate by bof, kingW3, Stefan Mesken, pjs36, Davide Giraudo May 31 '17 at 20:09

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  1. Total edges = $m.n$
  2. Avg = $\frac{2mn}{m+n}$
    Because total nodes =$m+n$, and $m$ nodes have $n$ degree, and $n$ nodes have $m$ degree.
  3. For Euler Circuit, both $m,n$ must be even.
    Because if each node has even degree, and its connected graph, Euler circuit exists.

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