# Bipartite graph with vertices partitioned

Let $G$ be a bipartite graph $G(V,E)$ with vertices partitioned into sets $V_1$ and $V_2$ such every vertex in $V_1$ has the same degree and every vertex in $V_2$ has the same degree.

How do I prove that the $\frac{\deg V_1}{\deg V_2}$ = $\frac{|V_2|}{|V_1|}$?

You should count the number of edges in two different ways. Assume the vertices in $V_1$ have degree $a$ and the vertices in $V_2$ have degree $b$. Then if we count the edges leaving $V_1$, well there are $a$ edges leaving each vertex, so there are $a\cdot |V_1|$ edges all together. Similarly if we count the edges by seeing how many leave $V_2$, we get that there are $b\cdot |V_2|$ edges. Thus $$a\cdot |V_1|=b\cdot |V_2|.$$ Thus we get the desired result.