On near optimal colorable graphs

TL;DR

This paper proves that certain ($F, K_4-e$)-free graphs are near optimal colorable, meaning their chromatic number is bounded by a constant or their clique number, and uses this to show the polynomial-time solvability of the coloring problem for these graphs.

Contribution

It establishes that ($F, K_4-e$)-free graphs are near optimal colorable, partially answering open questions and providing an alternative proof for polynomial-time coloring.

Findings

($F, K_4-e$)-free graphs are near optimal colorable

Chromatic number problem is polynomial-time solvable for these graphs

Provides partial answers to open questions in graph coloring

Abstract

A class of graphs $\cal G$ is said to be \emph{near optimal colorable} if there exists a constant $c\in \mathbb{N}$ such that every graph $G\in \cal G$ satisfies $\chi(G) \leq \max\{c, \omega(G)\}$, where $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. The class of near optimal colorable graphs is an important subclass of the class of $\chi$-bounded graphs which is well-studied in the literature. In this paper, we show that the class of ($F, K_4-e$)-free graphs is near optimal colorable, where $F\in \{P_1+2P_2,2P_1+P_3,3P_1+P_2\}$ and the graph $K_4-e$ is commonly referred as the {\em diamond}. This partially answers a question of Ju and Huang [Theoretical Computer Science 993 (2024) Article No.: 114465] and is related to a question of Schiermeyer (unpublished). Furthermore, using these results with some earlier known results, we also provide an alternate proof to the fact that the \textsc{Chromatic Number} problem for the class of ($F, K_4-e$)-free graphs is solvable in polynomial time, where $F\in \{P_1+2P_2,2P_1+P_3,3P_1+P_2\}$.