Let $2\leq{m}\leq{n}$ be an integer.
We denote the complete bipartite graph as $K_m,_n$
(a)How many circuits of length 4 contain $K_m,_n$?
(b)How long is the longest circuit in $K_m,_n$?
A bipartite graph, where m is the number of vertices in A and n is the number vertices in B.
A useful characterization of bipartite graphs: A graph G is bipartite if and only if you do not have circuits with an odd number of edges. => If G is bipartite with corners classes A and B, is every other corner in a circuit in A and every other is B. Thus, a circuit having an even number of corners.
in (a) i think for example m=3 and n=5 or m=4 and n=6 (Must be even number of corners). For a circuit of length 4 we will use 2 vertices in A and 2 vertices in B. My best guess, something like: $\binom{m}{2}$*$\binom{n}{2}$
Can someone help me with (a) and (b)?