Optimal chromatic bound for ($P_2+P_3$, $\bar{P_2+ P_3}$)-free graphs

TL;DR

This paper establishes the exact upper bound on the chromatic number for ($P_2+P_3$, $ar{P_2+P_3}$)-free graphs, showing the bound is tight and depends on the clique number.

Contribution

It provides the first tight bound on the chromatic number for this class of graphs, linking it explicitly to the clique number.

Findings

The chromatic number is at most max{ω+3, floor(3ω/2)-1} for these graphs.

The bound is tight, with constructions matching the upper bound.

The result advances understanding of coloring in graph classes defined by forbidden subgraphs.

Abstract

For a graph $G$, let $\chi(G)$ ($\omega(G)$) denote its chromatic (clique) number. A $P_2+P_3$ is the graph obtained by taking the disjoint union of a two-vertex path $P_2$ and a three-vertex path $P_3$. A $\bar{P_2+P_3}$ is the complement graph of a $P_2+P_3$. In this paper, we study the class of ($P_2+P_3$, $\bar{P_2+P_3}$)-free graphs and show that every such graph $G$ with $\omega(G)\geq 3$ satisfies $\chi(G)\leq \max \{\omega(G)+3, \lfloor\frac{3}{2} \omega(G) \rfloor-1 \}$. Moreover, the bound is tight. Indeed, for any $k\in {\mathbb N}$ and $k\geq 3$, there is a ($P_2+P_3$, $\bar{P_2+P_3}$)-free graph $G$ such that $\omega(G)=k$ and $\chi(G)=\max\{k+3, \lfloor\frac{3}{2} k \rfloor-1 \}$.