What are the number of 4 cycles in the Complete Bipartite graph? The question I have is:  How many cycles of Length 4 are there in a Complete Bipartite graph in $K_{n,n}$?  I can see there is 1 distinct cycle in $K_{2,2}$ but after that I can't seem to get it straight.  I think I can see 4 distinct 4 cycles in $K_{3,3}$ but am not sure as I keep losing track of which I have counted.  Thus I can't get past that to see a relationship and work from there.  Any ideas?
Thanks,
Brian

HINT: Let $V_0$ and $V_1$ be the two parts of $K_{n,n}$, so that each consists of $n$ vertices. Every $4$ cycle contains two vertices from $V_0$ and two from $V_1$. Suppose that you pick $v_0,v_1\in V_0$ and $u_0,u_1\in V_1$, where $v_0\ne v_1$ and $u_0\ne u_1$. They and their connecting edges form a little $K_{2,2}$, which indeed has one $4$-cycle. Thus, $K_{n,n}$ has exactly as many $4$-cycles as there are ways to pick two vertices from $V_0$ and two from $V_1$.


*

*How many ways are there to choose two vertices from the $n$ in $V_0$?  

*How many ways are there to choose two vertices from the $n$ in $V_0$?  

*How do you combine those two numbers to get the answer?

