Coloring ($P_5$, kite)-free graphs

TL;DR

This paper investigates the chromatic number of ($P_5$, kite)-free graphs with small clique numbers, establishing tight upper bounds for graphs excluding larger cliques, and providing examples to demonstrate these bounds are optimal.

Contribution

It extends known bounds on chromatic number for ($P_5$, kite)-free graphs by determining tight bounds for graphs excluding $K_6$ and $K_7$, and constructs examples showing these bounds are sharp.

Findings

For ($P_5$, kite, $K_6$)-free graphs, $oxed{ ext{chromatic number} oxed{ ext{} ext{up to } 7}$]

For ($P_5$, kite, $K_7$)-free graphs, $oxed{ ext{chromatic number} oxed{ ext{} ext{up to } 9}$]

Bounds are proven to be tight with explicit examples.

Abstract

Let $P_n$ and $K_n$ denote the induced path and complete graph on $n$ vertices, respectively. The {\em kite} is the graph obtained from a $P_4$ by adding a vertex and making it adjacent to all vertices in the $P_4$ except one vertex with degree 1. A graph is ($P_5$, kite)-free if it has no induced subgraph isomorphic to a $P_5$ or a kite. For a graph $G$, the chromatic number of $G$ (denoted by $\chi(G)$) is the minimum number of colors needed to color the vertices of $G$ such that no two adjacent vertices receive the same color, and the clique number of $G$ is the size of a largest clique in $G$. Here, we are interested in the class of ($P_5$, kite)-free graphs with small clique number. It is known that every ($P_5$,~kite, $K_3$)-free graph $G$ satisfies $\chi(G)\leq 3$, every ($P_5$,~kite, $K_4$)-free graph $G$ satisfies $\chi(G)\leq 4$, and that every ($P_5$,~kite, $K_5$)-free graph $G$ satisfies $\chi(G)\leq 6$. In this paper, we showed the following: $\bullet$ Every ($P_5$, kite, $K_6$)-free graph $G$ satisfies $\chi(G)\leq 7$. $\bullet$ Every ($P_5$, kite, $K_7$)-free graph $G$ satisfies $\chi(G)\leq 9$. We also give examples to show that the above bounds are tight.