A bound on the chromatic number using the longest odd cycle length

  • Authors:

  • Sreyash Kenkre;Sundar Vishwanathan

  • Affiliations:

  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai-400076, India;Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai-400076, India

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2007

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Abstract

Let G be a non-bipartite graph with 𝓵 as thelength of the longest odd cycle. Erdös and Hajnal proved thatχ(G) ≤ 𝓵 + 1. We show that the only graphsfor which this is tight are those that containK𝓵 + 1 and further, if G does notcontain K𝓵 then χ(G) ≤𝓵 -1. We then extend these results and show that if weexclude a large clique, we can prove better bounds forχ(G) as a function of 𝓵. Specifically, we showthat if ω(G) ≤ 𝓵(1 - ε) forε ≤ 4-15, then χ(G) ≤ 𝓵(1 -ε-2) . © 2006 Wiley Periodicals, Inc. J Graph Theory54: 267276, 2007