Near optimal colourability on (H, Kn−e)-free graphs

Abstract

A graph family $G$ is near optimal colourable if there is a constant number $c$, such that every graph $G\in G$ satisfies $\chi \left(G\right)\le max\{c,\omega \left(G\right)\}$, where $\chi \left(G\right)$ and $\omega \left(G\right)$ are the chromatic number and clique number of $G$, respectively. One may reduce the colouring problem on a near optimal colourable graph family to $q$-colouring problems for $q\le c-1$. In our previous paper [Y. Ju and S. Huang. Near optimal colourability on hereditary graph families. Theoretical Computer Science 994: 114465, 2024], we give an almost complete characterization for the near optimal colourability for ($H_1,H_2$)-free graphs and give the open problem: “Decide whether the family of ($H_1,H_2$)-free graphs is near optimal colourable, when $H_1$ is a forest with independence number at least 3 and $H_2=K_n-e$ ($n\ge 4$).” In this paper, we partially solve this open problem. We prove that for every $n\ge 4$, the family of ($H$, $K_n-e$)-free graphs is near optimal colourable if $H$ is an induced subgraph of $P_5$, $P_3+P_2$ or $m{P}_1+P_2$ for any $m$, and not near optimal colourable if $H$ contains an induced claw. Using these results, we give a complete classification on the near optimal colourability of ($H$, $K_n-e$)-free graphs, when $\left|H\right|\le 4$: the family of ($H$, $K_n-e$)-free graphs ($\left|H\right|\le 4$, $n\ge 4$) is near optimal colourable if and only if $H$ is a linear forest.