New results for MaxCut in H$H$‐free graphs

Abstract

AbstractThe MaxCut problem asks for the size of a largest cut in a graph . It is well known that for any ‐edge graph , and the difference is called the surplus of . The study of the surplus of ‐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 , and found constructions matching this bound. We prove several new results in this area. Confirming a conjecture of Alon, Krivelevich and Sudakov, we show that for every fixed odd , any ‐free graph with edges has surplus . This is tight, as is shown by a construction of pseudo‐random ‐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 is even. Generalising 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. We prove that an ‐vertex graph with few copies of and average degree has surplus , which is tight when is close to provided that a conjectured dense pseudo‐random ‐free graph exists. This result is used to improve the best known lower bound (as a function of ) on the surplus of ‐free graphs. Our proofs combine techniques from semi‐definite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.