Forbidden induced pairs for perfectness andω -colourability of graphs

• Main Authors: Chudnovsky, Maria, Kabela, Adam, Li, Binlong, Vrána, Petr
• Format: Journal Article
• Language: English
• Published: 16.08.2021
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2108.07071

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.