A superlocal version of Reed's Conjecture

TL;DR

This paper introduces a new local strengthening of Reed's conjecture on graph chromatic number, considering neighborhoods of two adjacent vertices, and provides evidence supporting its validity in specific graph classes.

Contribution

It proposes a novel local bound based on neighborhoods of two adjacent vertices and demonstrates its validity in fractional relaxations, quasi-line graphs, and graphs with stability number two.

Findings

The stronger bound holds in the fractional relaxation.

The bound is valid for quasi-line graphs.

The bound applies to graphs with stability number two.

Abstract

Reed's well-known $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi \leq \lceil \frac 12(\Delta+1+\omega)\rceil$. The second author formulated a {\em local strengthening} of this conjecture that considers a bound supplied by the neighbourhood of a single vertex. Following the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that considers a bound supplied by the neighbourhoods of two adjacent vertices. We provide some fundamental evidence in support, namely that the stronger bound holds in the fractional relaxation and holds for both quasi-line graphs and graphs with stability number two. We also conjecture that in the fractional version, we can push the locality even further.