# Complete Bipartite Graph. Define $K_{mn}$ [duplicate]

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 GiraudoMay 31 '17 at 20:09

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.