Max-Cut in Degenerate $H$-Free Graphs

TL;DR

This paper establishes new lower bounds on the maximum cut size in $d$-degenerate $H$-free graphs, generalizing recent results and connecting to longstanding conjectures in graph theory.

Contribution

It provides generalized bounds on Max-Cut in $d$-degenerate $H$-free graphs, extending previous work and proposing a conjecture linking Max-Cut bounds to the ABKS conjecture.

Findings

Derived lower bounds for Max-Cut in $H$-free graphs.

Extended bounds for cycles, odd wheels, and bipartite graphs.

Connected Max-Cut bounds to the ABKS conjecture.

Abstract

We obtain several lower bounds on the $\textsf{Max-Cut}$ of $d$-degenerate $H$-free graphs. Let $f(m,d,H)$ denote the smallest $\textsf{Max-Cut}$ of an $H$-free $d$-degenerate graph on $m$ edges. We show that $f(m,d,K_r)\ge \left(\frac{1}{2} + d^{-1+\Omega(r^{-1})}\right)m$, generalizing a recent work of Carlson, Kolla, and Trevisan. We also give bounds on $f(m,d,H)$ when $H$ is a cycle, odd wheel, or a complete bipartite graph with at most 4 vertices on one side. We also show stronger bounds on $f(m,d,K_r)$ assuming a conjecture of Alon, Bollabas, Krivelevich, and Sudakov (2003). We conjecture that $f(m,d,K_r)= \left( \frac{1}{2} + \Theta_r(d^{-1/2}) \right)m$ for every $r\ge 3$, and show that this conjecture implies the ABKS conjecture.