On hitting all maximum cliques with an independent set

TL;DR

This paper establishes conditions under which graphs contain independent sets reducing their maximum clique size, providing insights into Reed's conjecture and bounds on chromatic number related to maximum degree and clique size.

Contribution

It proves a new bound linking maximum clique size and maximum degree, and offers a generic proof for an upper bound on chromatic number for line graphs of multigraphs.

Findings

Graphs with clique number at least 3/4 of maximum degree plus one have independent sets reducing the clique number.

Minimum counterexamples to Reed's conjecture must have clique number less than 3/4 of maximum degree plus one.

A new upper bound on chromatic number based on local properties of induced subgraphs.

Abstract

We prove that every graph $G$ for which $\omega(G) \geq 3/4(\Delta(G) + 1)$, has an independent set $I$ such that $\omega(G - I) < \omega(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $\omega(G) < 3/4(\Delta(G) + 1)$ and hence also $\chi(G) > \lceil 7/6\omega(G) \rceil$. We also prove that if for every induced subgraph $H$ of $G$ we have $\chi(H) \leq \max{\lceil 7/6\omega(H) \rceil, \lceil \frac{\omega(H) + \Delta(H) + 1}{2}\rceil}$, then we also have $\chi(G) \leq \lceil \frac{\omega(G) + \Delta(G) + 1}{2}\rceil$. This gives a generic proof of the upper bound for line graphs of multigraphs proved by King et al.