Near optimal colourability on hereditary graph families - Ju, Yiao; Huang, Shenwei (Theoretical computer science, 2024)
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 Lovász theta function. A graph family G is near optimal colourable if there is a constant number c such that every graph G∈G satisfies χ(G)≤max⁡{c,ω(G)}, where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. The near optimal colourable graph families together with the Lovász theta function are useful for the study of the colouring problems for hereditary graph families. We investigate the near optimal colourability for (H1,H2)-free graphs. Our main result is an almost complete characterisation for the near optimal colourability for (H1,H2)-free graphs with two exceptional cases, one of which is the celebrated Gyárfás conjecture. As an application of our results, we show that the colouring problem for (2K2,P4∨Kn)-free graphs is polynomial time solvable, which solves an open problem in Dabrowski and Paulusma (2018) [6].