Chromatic Number, Induced Cycles, and Non-separating Cycles

Abstract

We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge contraction, we obtain new upper bounds on the chromatic number in terms of the number of induced cycles and the Hadwiger number in terms of the number of induced non-separating cycles. As an application, we show that every 3-connected graph with average degree at least 2k for some \(k\ge 2\) have at least \((k-1)|V|+Ck^{3}\log ^{3/2}k\) induced non-separating cycles for some explicit constant \(C>0\). This improves the previous best known lower bound \((k-1)|V|+1\), which follows from Tutte’s cycle space theorem. We also give a short proof of this theorem of Tutte.

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Acknowledgements

The author is grateful for the anonymous referees for their valuable suggestions, which helped improving the presentation significantly. The author also thanks to Neil Robertson and Paul Seymour for helpful discussions.

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Correspondence to Hanbaek Lyu.

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Lyu, H. Chromatic Number, Induced Cycles, and Non-separating Cycles. Graphs and Combinatorics 36, 1297–1310 (2020). https://doi.org/10.1007/s00373-020-02187-4

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Keywords

  • Chromatic number
  • Induced cycles
  • Hadwiger number
  • Induced non-separating cycles
  • Euler characteristic

Mathematics Subject Classification

  • 05C15
  • 05C83
  • 05C38