Reed's conjecture on some special classes of graphs

TL;DR

This paper verifies Reed's conjecture for specific classes of graphs, including subclasses of P5-free and Chair-free graphs, expanding the understanding of graph coloring bounds in these categories.

Contribution

The paper proves Reed's conjecture holds for certain special graph classes, notably subclasses of P5-free and Chair-free graphs, which were previously unverified.

Findings

Reed's conjecture is valid for subclasses of P5-free graphs.

Reed's conjecture is valid for subclasses of Chair-free graphs.

The results extend the classes of graphs known to satisfy Reed's conjecture.

Abstract

Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G)+\Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$. In this paper, we verify this conjecture for some special classes of graphs, in particular for subclasses of $P_5$-free graphs or $Chair$-free graphs.