Pure Pairs. VIII. Excluding a Sparse Graph.

Abstract

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.