New results for MaxCut in $H$-free graphs

Abstract

The MaxCut problem asks for the size mc(G) of a largest cut in a graph G. It is well known that mc(G) ≥ m/2 for any m-edge graph G, and the difference mc(G) − m/2 is called the surplus of G. The study of the surplus of H-free graphs was initiated by Erdős and Lovász in the 70s, who in particular asked what happens for triangle-free graphs. This was famously resolved by Alon, who showed that in the triangle-free case the surplus is Ω(m 4/5 ), and found constructions matching this bound. We prove several new results in this area.(i) We show that for every fixed odd r ≥ 3, any C r -free graph with m edges has surplus Ω r m r+1 r+2 . This is tight, as is shown by a construction of pseudorandom C r -free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when r is even.(ii) Generalizing the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs our bounds on the surplus are sharp.(iii) We prove that an n-vertex graph with few copies of K r and average degree d has surplus Ω r (d r−1 /n r−3 ), which is tight when d is close to n provided that a conjectured dense pseudorandom K r -free graph exists. This result is used to improve the best known lower bound (as a function of m) on the surplus of K r -free graphs.Our proofs combine techniques from semidefinite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.

Max-Cut by Excluding Bipartite Subgraphs

“…Recently, Lin [19] improved (1.1) by showing that f (m, Θ k,t ) ≥ m/2 + Ω(m (2k+1)/(2k+2) ). For more problems and results on f (m, H), see [15,16,20,23].…”