A tight linear bound to the chromatic number of $(P_5, K_1+(K_1\cup K_3))$-free graphs

TL;DR

This paper characterizes certain graph classes and establishes tight bounds on their chromatic number relative to their clique number, advancing understanding of coloring properties in complex graph families.

Contribution

It provides a characterization of $(P_5, K_1igcup K_3)$-free graphs and proves a tight linear bound on their chromatic number in terms of their clique number.

Findings

Proves $oxed{ ext{chi}(G) ext{ } extless= 2 ext{ } ext{omega}(G)-1}$ for $(P_5, K_1igcup K_3)$-free graphs.

Establishes $oxed{ ext{chi}(G) extless= ext{max}igrace{2 ext{ } ext{omega}(G), 15}igrace}$ for $(P_5, K_1+(K_1igcup K_3))$-free graphs.

Constructs an infinite family of graphs where $ ext{chi}(G)=2 ext{ } ext{omega}(G)$.

Abstract

Let $F_1$ and $F_2$ be two disjoint graphs. The union $F_1\cup F_2$ is a graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)$, and the join $F_1+F_2$ is a graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)\cup \{xy\;|\; x\in V(F_1)\mbox{ and } y\in V(F_2)\}$. In this paper, we present a characterization to $(P_5, K_1\cup K_3)$-free graphs, prove that $\chi(G)\le 2\omega(G)-1$ if $G$ is $(P_5, K_1\cup K_3)$-free. Based on this result, we further prove that $\chi(G)\le $max$\{2\omega(G),15\}$ if $G$ is a $(P_5,K_1+( K_1\cup K_3))$-free graph, and construct an infinite family of $(P_5, K_1+( K_1\cup K_3))$-free graphs such that every graph $G$ in the family satisfies $\chi(G)=2\omega(G)$.