A Strengthening on Odd Cycles in Graphs of Given Chromatic Number

  title={A Strengthening on Odd Cycles in Graphs of Given Chromatic Number},
  author={Jun-ming Gao and Qingyi Huo and Jie Ma},
  journal={SIAM Journal on Discrete Mathematics},
Resolving a conjecture of Bollobás and Erdős, Gyárfás proved that every graph G of chromatic number k + 1 ≥ 3 contains cycles of ⌊ 2 ⌋ distinct odd lengths. We strengthen this prominent result by showing that such G contains cycles of ⌊ 2 ⌋ consecutive odd lengths. Along the way, combining extremal and structural tools, we prove a stronger statement that every graph of chromatic number k + 1 ≥ 7 contains k cycles of consecutive lengths, except that some block is Kk+1. As corollaries, this… 

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