Maximum bisections of graphs without cycles of length four and five

A bisection of a graph is a bipartition of its vertex set in which the two parts differ in size by at most 1, and its size is the number of edges which across the two parts. Let $G$ be a graph with $n$ vertices, $m$ edges and degree sequence $d_1,d_2,\dots ,d_n$. Motivated by a few classical results on Max-Cut of graphs, Lin and Zeng proved that if $G$ is $\{C_4,C_6\}$-free and has a perfect matching, then $G$ has a bisection of size at least $m/2+\Omega \left({\sum }_{i=1}^n\sqrt{d_i}\right)$, and conjectured the same bound holds for $C_4$-free graphs with perfect matchings. In this paper, we confirm the conjecture under the additional condition that $G$ is $C_5$-free.