The chromatic number of ($P_{5}, K_{5}-e$)-free graphs

TL;DR

This paper proves that graphs free of both a path on five vertices and a nearly complete five-clique have their chromatic number bounded linearly by their clique number, advancing understanding of graph coloring constraints.

Contribution

It establishes that the family of (P5, K5-e)-free graphs is χ-bounded by a linear function, specifically χ(G) ≤ max{13, ω(G)+1}.

Findings

The family of (P5, K5-e)-free graphs is χ-bounded by a linear function.

The chromatic number is at most ω(G)+1 for these graphs.

A universal bound of 13 applies when the clique number is small.

Abstract

Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices. A family of graphs $\mathcal{G}$ is said to be {\it$\chi$-bounded} if there exists some function $f$ such that $\chi(G)\leq f(\omega(G))$ for every $G\in\mathcal{G}$. In this paper, we show that the family of $(P_5, K_5-e)$-free graphs is $\chi$-bounded by a linear function: $\chi(G)\leq \max\{13,\omega(G)+1\}$.