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arXiv · 2205.08291

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

Wei Dong·Baogang Xu·Yian Xu
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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 $\chi(G)=2\omega(G)$.

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Wei Dong, Baogang Xu, Yian Xu. 2022-05-17. A tight linear bound to the chromatic number of $(P_5, K_1+(K_1\cup K_3))$-free graphs. https://arxiv.org/abs/2205.08291

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