Prove that if a $k$-regular bipartite graph has a bipartition $(x,y)$ then $\vert x\vert=\vert y \vert$

Question

A problem I don't know how to attack... a bipartite is supposed to have one end in $x$ and one in $y$. A graph is $k$-regular if $d(v)=k$ for all $v\in v(G)$

Answer 1

Number of edges in graph = $k|x| = k|y|$. Now divide by k.

Answer 2

Since $G$ is bipartite we can split the vertex set of $G$ into two sets, say $x$ and $y$. We also know that each edge of $G$ has one end in $x$ and one end in $y$. So since $G$ is $k$-regular we have $k|x|=k|y|$ edges in $G$ which implies that $|x|=|y|$.

Answer 3

Use the technique of double-counting: count the number of edges in the graph in two different ways - by adding the degrees of all vertices in the left subset $x$, and then by adding the degrees of all vertices in the right subset $y$. These two sums must be equal since they count the same set of edges.