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
P. Erdös, Problems and results in graph theory and combinatorial analysis, inGraph Theory and Related Topics, Academic Press, New York, 1979, 153–163.
L. Lovász,unpublished.
J. Schmerl, Recursive colorings of graphs,Can. J. Math.,XXXII. 4 (1980) 821–830.
A. Schrijver, Vertex-critical subgraphs of Kneser graphs,reprint, Amsterdam (1978).
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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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DOI: https://doi.org/10.1007/BF02579219