Near Optimal Colourability on Hereditary Graph Families

In this paper, we initiate a systematic study on a new notion called near optimal colourability which is closely related to perfect graphs and the Lov{\'a}sz theta function. A graph family $\mathcal{G}$ is {\em near optimal colourable} if there is a constant number $c$ such that every graph $G\in\mathcal{G}$ satisfies $\chi(G)\leq\max\{c, \omega(G)\}$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. The near optimal colourable graph families together with the Lov{\'a}sz theta function are useful for the study of the chromatic number problems for hereditary graph families. We investigate the near optimal colourability for ($H_1,H_2$)-free graphs. Our main result is an almost complete characterization for the near optimal colourability for ($H_1,H_2$)-free graphs with two exceptional cases, one of which is the celebrated Gy{\'a}rf{\'a}s conjecture. As an application of our results, we show that the chromatic number problem for ($2K_2,P_4\vee K_n$)-free graphs is polynomial time solvable, which solves an open problem in [K.~K.~Dabrowski and D.~Paulusma. On colouring ($2P_2$, $H$)-free and ($P_5$, $H$)-free graphs. Information Processing Letters, 134:35-41, 2018].