Forbidden induced pairs for perfectness and $\omega$-colourability of graphs

TL;DR

This paper characterizes pairs of graphs that, when forbidden, ensure all resulting graphs are perfect or have chromatic number equal to their clique number, extending previous work with new structural insights.

Contribution

It provides new characterizations of forbidden graph pairs that guarantee perfectness and $oldsymbol{ ext{omega}}$-colourability for broad classes of graphs, including non-hereditary classes.

Findings

Characterizations of pairs $oxed{ ext{X,Y}}$ ensuring perfectness.

Characterizations of pairs $oxed{ ext{X,Y}}$ ensuring $oldsymbol{ ext{omega}}$-colourability.

Extensions to connected graphs with independence at least 3 and other constraints.

Abstract

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.