Excluding paths and antipaths

Abstract

The Erdős-Hajnal conjecture states that for every graph H, there exists a constant δ(H)>0, such that if a graph G has no induced subgraph isomorphic to H, then G contains a clique or a stable set of size at least |V (G)|δ(H). This conjecture is still open. We consider a variant of the conjecture, where instead of excluding H as an induced subgraph, both H and H c are excluded. We prove this modified conjecture for the case when H is the five-edge path. Our second main result is an asymmetric version of this: we prove that for every graph G such that G contains no induced six-edge path, and G c contains no induced four-edge path, G contains a polynomial-size clique or stable set.

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Correspondence to Maria Chudnovsky.

Additional information

Supported by NSF grants DMS-1001091 and IIS-1117631.

Supported by ONR grant N00014-10-1-0680 and NSF grant DMS-0901075.

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Chudnovsky, M., Seymour, P. Excluding paths and antipaths. Combinatorica 35, 389–412 (2015). https://doi.org/10.1007/s00493-014-3000-z

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Mathematics Subject Classification (2000)

  • 05C38
  • 05C75