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

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 $|x|=|y|$.
