On coloring graphs with locally small chromatic number

Abstract

In 1973, P. Erdös conjectured that for eachkε2, there exists a constantc k so that ifG is a graph onn vertices andG has no odd cycle with length less thanc k n 1/k, then the chromatic number ofG is at mostk+1. Constructions due to Lovász and Schriver show thatc k , if it exists, must be at least 1. In this paper we settle Erdös’ conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.

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References

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Research supported by NSF Grant ISP-8 011 451.

Research supported by NSF Grant MCS-8 202 172.

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Kierstead, H.A., Szemerédi, E. & Trotter, W.T. On coloring graphs with locally small chromatic number. Combinatorica 4, 183–185 (1984). https://doi.org/10.1007/BF02579219

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AMS subject classification (1980)

  • 05 C 15