1. ## Complete Bipartite Graphs

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

(a)
Prove That

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

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

2. Recall that if $v\in V_1$ then $\text{Deg}(v)=\left|V_2\right|$.
So part a follows from that.

Part b is from the definition of regularity.