Forbidden induced pairs for perfectness and $ω$-colourability of graphs

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 $ω$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $ω$-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 $ω$-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 $χ$-boundedness and deciding $k$-colourability in polynomial time.

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