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A New Upper Bound on the Chromatic Number of Graphs with No Odd Kt Minor

Abstract

Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colorable. This is a strengthening of the famous Hadwiger’s Conjecture. Geelen et al. proved that every graph with no odd Kt minor is \(O(t\sqrt {\log t} )\)-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no Kt minor, we make the first improvement on this bound by showing that every graph with no odd Kt minor is O(t(logt)β)-colorable for every β > 1/4.

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Acknowledgements

The authors would like to thank referees for their valuable comments. The research presented in this paper was partially conducted during the visit of the second author to the Institute of Basic Science in Daejeon, Korea. The second author thanks the Institute of Basic Science for the hospitality, and Sang-il Oum for helpful discussion.

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Correspondence to Zi-Xia Song.

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Supported by an NSERC grant.

Supported by the National Science Foundation under Grant No. DMS-1854903.

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Norin, S., Song, ZX. A New Upper Bound on the Chromatic Number of Graphs with No Odd Kt Minor. Combinatorica 42, 137–149 (2022). https://doi.org/10.1007/s00493-021-4390-3

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  • DOI: https://doi.org/10.1007/s00493-021-4390-3

Mathematics Subject Classification (2010)

  • 05C15
  • 05C83