Lower bounds for Max-Cut in $H$-free graphs via semidefinite programming

TL;DR

This paper introduces a semidefinite programming approach to establish lower bounds on the maximum cut size in graphs, especially focusing on graphs with few triangles and those free of complete subgraphs.

Contribution

The paper develops a novel SDP-based method for deriving lower bounds on Max-Cut in specific classes of $H$-free graphs, extending previous techniques.

Findings

Effective lower bounds for Max-Cut in triangle-free graphs

New bounds for $K_r$-free graphs

Demonstrates the power of SDP in combinatorial optimization

Abstract

For a graph $G$, let $f(G)$ denote the size of the maximum cut in $G$. The problem of estimating $f(G)$ as a function of the number of vertices and edges of $G$ has a long history and was extensively studied in the last fifty years. In this paper we propose an approach, based on semidefinite programming (SDP), to prove lower bounds on $f(G)$. We use this approach to find large cuts in graphs with few triangles and in $K_r$-free graphs.