Infinitely connected subgraphs in graphs of uncountable chromatic number

Abstract

Erdős and Hajnal conjectured in 1966 that every graph of uncountable chromatic number contains a subgraph of infinite connectivity. We prove that every graph of uncountable chromatic number has a subgraph which has uncountable chromatic number and infinite edge-connectivity. We also prove that, if each orientation of a graph G has a vertex of infinite outdegree, then G contains an uncountable subgraph of infinite edge-connectivity.

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Correspondence to Carsten Thomassen.

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Thomassen, C. Infinitely connected subgraphs in graphs of uncountable chromatic number. Combinatorica 37, 785–793 (2017). https://doi.org/10.1007/s00493-016-3436-4

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C40
  • 05C63