An optimal chromatic bound for ($P_2+P_3$, gem)-free graphs

TL;DR

This paper determines the exact smallest chi-binding function for the class of ($P_2+ P_3$, gem)-free graphs, establishing a precise bound on the chromatic number based on the clique number.

Contribution

It introduces the first exact smallest chi-binding function for ($P_2+ P_3$, gem)-free graphs, advancing understanding of chromatic bounds in this graph class.

Findings

The function $oxed{ ext{phi}(x)=1, 4, 6, ext{ and } ig\lceilrac{1}{4}(5x-1)ig ceil ext{ for } x ext{ ≥ 4}}$ is the smallest chi-binding function.

Proves that for ($P_2+ P_3$, gem)-free graphs, the chromatic number is bounded by phi of the clique number.

Establishes the tightness of the bound, showing it cannot be improved.

Abstract

Given a graph $G$, the parameters $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and the clique number of $G$. A function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(1) = 1$ and $f(x) \geq x$, for all $x \in \mathbb{N}$ is called a $\chi$-binding function for the given class of graphs $\cal{G}$ if every $G \in \cal{G}$ satisfies $\chi(G) \leq f(\omega(G))$, and the \emph{smallest $\chi$-binding function} $f^*$ for $\cal{G}$ is defined as $f^*(x) := \max\{\chi(G)\mid G\in {\cal G} \mbox{ and } \omega(G)=x\}$. In general, the problem of obtaining the smallest $\chi$-binding function for the given class of graphs seems to be extremely hard, and only a few classes of graphs are studied in this direction. In this paper, we study the class of ($P_2+ P_3$, gem)-free graphs, and prove that the function $\phi:\mathbb{N}\rightarrow \mathbb{N}$ defined by $\phi(1)=1$, $\phi(2)=4$, $\phi(3)=6$ and $\phi(x)=\left\lceil\frac{1}{4}(5x-1)\right\rceil$, for $x\geq 4$ is the smallest $\chi$-binding function for the class of ($P_2+ P_3$, gem)-free graphs.