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

Abstract

The MaxCut problem asks for the size mcfalse(Gfalse)${\rm mc}(G)$ of a largest cut in a graph G$G$. It is well known that mcfalse(Gfalse)⩾m/2${\rm mc}(G)\geqslant m/2$ for any m$m$‐edge graph G$G$, and the difference mcfalse(Gfalse)−m/2${\rm mc}(G)-m/2$ is called the surplus of G$G$. The study of the surplus of H$H$‐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 normalΩfalse(m4/5false)$\Omega (m^{4/5})$, and found constructions matching this bound. We prove several new results in this area. (i)Confirming a conjecture of Alon, Krivelevich and Sudakov, we show that for every fixed odd r⩾3$r\geqslant 3$, any Cr$C_r$‐free graph with m$m$ edges has surplus Ωr(mr+1r+2)$\Omega _r(m^{\frac{r+1}{r+2}})$. This is tight, as is shown by a construction of pseudo‐random Cr$C_r$‐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$r$ is even. (ii)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. (iii)We prove that an n$n$‐vertex graph with few copies of Kr$K_r$ and average degree d$d$ has surplus Ωr(dr−1/nr−3)$\Omega _r(d^{r-1}/n^{r-3})$, which is tight when d$d$ is close to n$n$ provided that a conjectured dense pseudo‐random Kr$K_r$‐free graph exists. This result is used to improve the best known lower bound (as a function of m$m$) on the surplus of Kr$K_r$‐free graphs. Our proofs combine techniques from semi‐definite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.