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) ≥ m/2 + c(H)m^4/5, and that this is tight up to the value of $c(H)$. We also prove that for any even cycle $C_{2k}$ there is a c(k)> 0 such that f(m,C_2k) ≥ m/2 + c(k) m^{(2k+1)/(2k+2)}, and that this is tight, up to the value of $c(k)$, for 2k ∈ {4,6,10\}. The proofs combine combinatorial, probabilistic and spectral techniques.