Let G be a bipartite graph with vertex partition {$\displaystyle V_1, V_2$}.

a)Prove that:

$\displaystyle \sum_{x \exists V_1}\deg(x) = |E| = \sum_{y \exists V_2}\deg(y) $


b) A graph is $\displaystyle k-regular$ if every vertex has degree $\displaystyle k$. Prove that if $\displaystyle G$ is $\displaystyle k-regular $ with $\displaystyle k > 0 $, then $\displaystyle |V_1| = |V_2| $ (NOTE: G is bipartite with vertex partition {$\displaystyle V_1, V_2$}.)