Browsing by Author "Seymour, Paul"

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Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path.
(Springer, 2023-09-15) Scott, Alex; Seymour, Paul; Spirkl, Sophie
A graph G is H -free if it has no induced subgraph isomorphic to H. We prove that a P5-free graph with clique number ω ≥ 3 has chromatic number at most ωlog2(ω). The best previous result was an exponential upper bound (5/27)3ω, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erd˝os-Hajnal conjecture holds for P5, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P5-free graphs, and our result is an attempt to approach that.
Pure Pairs. VIII. Excluding a Sparse Graph.
(Springer, 2024-08-05) Scott, Alex; Seymour, Paul; Spirkl, Sophie
A pure pair of size t in a graph G is a pair A, B of disjoint subsets of V(G), each of cardinality at least t, such that A is either complete or anticomplete to B. It is known that, for every forest H, every graph on n ≥ 2 vertices that does not contain H or its complement as an induced subgraph has a pure pair of size (n); furthermore, this only holds when H or its complement is a forest. In this paper, we look at pure pairs of size n1−c, where 0 < c < 1. Let H be a graph: does every graph on n ≥ 2 vertices that does not contain H or its complement as an induced subgraph have a pure pair of size (|G| 1−c)? The answer is related to the congestion of H, the maximum of 1 − (|J | − 1)/|E(J )| over all subgraphs J of H with an edge. (Congestion is nonnegative, and equals zero exactly when H is a forest.) Let d be the smaller of the congestions of H and H. We show that the answer to the question above is “yes” if d ≤ c/(9 + 15c), and “no” if d > c.