Nearly bipartite graphs with large chromatic number

Abstract

P. Erdős and A. Hajnal asked the following question. Does there exist a constant ε>0 with the following property: If every subgraphH of a graphG can be made bipartite by the omission of at most ε|H| edges where |H| denotes the number of vertices ofH thenx(H) ≦ 3.

The aim of this note is to give a negative answer to this question and consider the analogous problem for hypergraphs. The first was done also by L. Lovász who used a different construction.

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Rödl, V. Nearly bipartite graphs with large chromatic number. Combinatorica 2, 377–383 (1982). https://doi.org/10.1007/BF02579434

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AMS subject classification (1980)

  • 05 C 15
  • 05 C 65