MaxCut in H-free graphs

For a graph G, let f(G) denote the maximum number of edges in a cut of G. For an integer m and for a fixed graph H, let f(m,H) denote the minimum possible cardinality of f(G), as G ranges over all graphs on m edges that contain no copy of H. In this paper we study this function for various graphs H. In particular we show that for any graph H obtained by connecting a single vertex to all vertices of a fixed nontrivial forest, there is a c(H) > 0 such that f(m,H) ≥ m2 + c(H)m , and this is tight up to the value of c(H). We also prove that for any even cycle C2k there is a c(k) > 0 such that f(m,C2k) ≥ m2 + c(k)m (2k+1)/(2k+2) and this is tight, up to the value of c(k), for 2k ∈ {4, 6, 10}. The proofs combine combinatorial, probabilistic and spectral techniques.