Maximum bipartite subgraphs in graphs without short cycles

Given a graph G, let f(G) denote the maximum number of edges in a bipartite subgraph of G. Given a set H of graphs and a positive integer m, let f (m, H ) denote the minimum possible cardinality of f (G), as G ranges over all graphs on m edges that contains no member of H as a subgraph. Suppose that r >= 10 is an even integer and k >= 2 is an integer. In this paper, we prove that there is a constant c(r) > 0 such that f (m, {C6, C7, ... , Cr-1}) >= m/2 + c(r)mr/(r+1) and there is a constant c(k) > 0 such that f (m, {C4, C6, . . . , C2k, C2k+1}) >= m/2 + c(k)m(2k+2)/(2k+3), both of which improve a result of Alon, Bollobas, Krivelevich and Sudakov. (c) 2021 Published by Elsevier B.V.