Forbidden Induced Pairs for Perfectness and $\omega$-Colourability of Graphs

• Published in: The Electronic journal of combinatorics Vol. 29; no. 2
• Main Authors: Chudnovsky, Maria, Kabela, Adam, Li, Binlong, Vrána, Petr
• Format: Journal Article
• Language: English
• Published: 06.05.2022
• ISSN: 1077-8926; 1097-1440; 1077-8926
• DOI: 10.37236/10708

We characterise the pairs of graphs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are perfect. Similarly, we characterise pairs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are $\omega$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $\omega$-colourability of all connected $\{ X, Y \}$-free graphs which are of independence at least~$3$, distinct from an odd cycle, and of order at least $n_0$, and similar characterisations subject to each subset of these additional constraints. (The classes are non-hereditary and the characterisations for perfectness and $\omega$-colourability are different.) We build on recent results of Brause et al. on $\{ K_{1,3}, Y \}$-free graphs, and we use Ramsey's Theorem and the Strong Perfect Graph Theorem as main tools. We relate the present characterisations to known results on forbidden pairs for $\chi$-boundedness and deciding $k$-colourability in polynomial time.