Coloring_of_some_crown-free_graphs

TL;DR

This paper establishes upper bounds on the chromatic number of crown-free graphs with additional forbidden subgraphs, advancing understanding of coloring properties in specific graph classes.

Contribution

It provides new upper bounds on the chromatic number for crown-free graphs excluding certain subgraphs, extending previous coloring theory results.

Findings

Proves $oxed{ ext{chromatic number} \\le rac{3}{2}( ext{clique number}^2 - ext{clique number})}$ for (crown, $P_5$)-free graphs.

Establishes $oxed{ ext{chromatic number} \\le rac{1}{2}( ext{clique number}^2 + ext{clique number})}$ for (crown, fork)-free graphs.

Shows $oxed{ ext{chromatic number} \\le rac{1}{2} ext{clique number}^2 + rac{3}{2} ext{clique number} + 1}$ for (crown, $P_3 old P_2$)-free graphs.

Abstract

Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup (H)$ and $E(G\cup H)=E(G)\cup E(H)$. The {\em join} $G+H$ is the graph with $V(G+H)=V(G)+V(H)$ and $E(G+H)=E(G)\cup E(H)\cup\{xy\;|\; x\in V(G), y\in V(H)$$\}$. We use $P_k$ to denote a {\em path} on $k$ vertices, use {\em fork} to denote the graph obtained from $K_{1,3}$ by subdividing an edge once, and use {\em crown} to denote the graph $K_1+K_{1,3}$. In this paper, we show that (\romannumeral 1) $\chi(G)\le\frac{3}{2}(\omega^2(G)-\omega(G))$ if $G$ is (crown, $P_5$)-free, (\romannumeral 2) $\chi(G)\le\frac{1}{2}(\omega^2(G)+\omega(G))$ if $G$ is (crown, fork)-free, and (\romannumeral 3) $\chi(G)\le\frac{1}{2}\omega^2(G)+\frac{3}{2}\omega(G)+1$ if $G$ is (crown, $P_3\cup P_2$)-free.