TITLE: Prove that if a $k$-regular bipartite graph has a bipartition $(x,y)$ then $\vert x\vert=\vert y \vert$
QUESTION [6 upvotes]: 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 $v\in v(G)$

REPLY [3 votes]: 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.