| Abstrakt: |
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 Ω (n) ; furthermore, this only holds when H or its complement is a forest. In this paper, we look at pure pairs of size n 1 - 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 Ω (| 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 + 15 c) , and "no" if d > c . [ABSTRACT FROM AUTHOR] |
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