Sharp lower bounds for bisections of graphs with bounded maximum degree

A bisection of a graph is a partition of its vertex set into two parts whose sizes differ by at most one. We prove that every graph with $m$ edges and maximum degree at most $k$ has a bisection of size at least $\frac{k+1}{2k}m$ when $k$ is odd, and at least $\frac{k+2}{2(k+1)}m$ when $k$ is even. B