Maximum bisections of graphs without cycles of length 4

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. Let C-k be a cycle of length k, and let G be a C-4-free graph with n vertices, m edges and vertex degrees d1, ... , dn. Lin and Zeng proved that if G does not contain C-6 and has a perfect matching, then G admits a bisection of size at least m/2+Omega(Sigma(n)(i=1) root d(i)). This extends a celebrated bound given by Shearer on Max-Cut of triangle-free graphs. In this paper, we establish a similar result by replacing C-6 with theta (1, 2, 4), theta (2, 3, 3) and theta (3, 3, 3), where theta(l(1), l(2), l(3)) denotes the graph consisting of three internally disjoint paths of length l(1), l(2) and l(3), respectively, each with the same endpoints. We also note that the bound is tight for certain polarity graphs. (c) 2022 Elsevier B.V. All rights reserved.