MaxCut in Graphs With Sparse Neighborhoods

Abstract

Let $G$ be a graph with $m$ edges and let $\text{mc}\left(G\right)$ denote the number of edges in a largest bipartite subgraph of $G$. The difference $\text{mc}\left(G\right)-m∕2$ is called the surplus $\text{sp}\left(G\right)$ of $G$. A fundamental problem in MaxCut is to determine $\text{sp}\left(G\right)$ for $G$ without specific structure, and the degree sequence $d_1,\dots ,d_n$ of $G$ plays a key role in this area. A classical example, given by Shearer, is that $\text{sp}\left(G\right)=\Omega \left({\sum }_{i=1}^n{\sqrt{d}}_i\right)$ for triangle-free graphs $G$. It was extended to graphs with sparse neighborhoods by Alon, Krivelevich and Sudakov. In this paper, we establish a novel and stronger result for a more general family of graphs with sparse neighborhoods. Our result derives many well-known bounds on surplus of $H$-free graphs for different $H$. Importantly, it can also deduce many new tight bounds when $H$ is any graph having a vertex whose removal results in a bipartite graph with Turán exponent at most 3/2, especially the even wheel. This contributes to a conjecture raised by Alon, Krivelevich and Sudakov.