Max-Bisections of graphs without perfect matching

A bisection of a graph is a bipartition of its vertex set such that the two resulting parts differ in size by at most 1, and its size is the number of edges that connect vertices in the two parts. The perfect matching condition and forbidden even cycles subgraphs are essential in finding large bisections of graphs. In this paper, we show that the perfect matching condition can be replaced by the minimum degree condition. Let $C_{\mathrm{\ell }}$ be a cycle of length $\mathrm{\ell }$ for $\mathrm{\ell }\ge 3$, and let $G$ be a $\{C_4,C_6\}$-free graph with $m$ edges and minimum degree at least 2. We prove that $G$ has a bisection of size at least $m\mathrm{/}2+\mathrm{\Omega }\left({\sum }_{v\in V(G)}\sqrt{d(v)}\right)$. As a corollary, if $G$ is also $C_{2k}$-free for $k\ge 3$, then $G$ has a bisection of size at least $m\mathrm{/}2+\mathrm{\Omega }\left(m^{(2k+1)\mathrm{/}(2k+2)}\right)$, thereby confirming a conjecture proposed by Lin and Zeng [J. Comb. Theory A, 180 (2021), 105404].