A Note Related to Graph Theory

TL;DR

This paper investigates the chromatic properties of a specific class of graphs that exclude certain subgraphs, establishing an upper bound on their chromatic number and applying this to related graph classes.

Contribution

It proves that $(P_3igcup P_2,K_4)$-free graphs have a chromatic number at most 7, providing new bounds for these and related graph classes.

Findings

$(P_3igcup P_2,K_4)$-free graphs are 7-colorable

Derived upper bounds for $(4K_1,igtriangleup P_3igcup P_2,K_{ ext{omega}})$-free graphs

Enhanced understanding of chromatic bounds in restricted graph classes

Abstract

This article foucuses on $(P_3\cup P_2,K_4)$-free graph. In this paper, we prove that if G is $(P_3\cup P_2,K_4)$-free, then $\chi(G)\le 7$. We then use our result to obtain the upper bound of order and chromatic number of $(4K_1,\overline{P_3\cup P_2},K_{\omega})$-free graph .