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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References
B. Bollobás and A. Thomason: Highly linked graphs, Combinatorica 16 (1996), 313–320.
P. A. Catlin: A bound on the chromatic number of a graph, Discrete Math. 22 (1978), 81–83.
M. Chudnovsky, J. Geelen, B. Gerards, L. Goddyn, M. Lohman and P. Seymour: Packing non-zero A-paths in group-labelled graphs, Combinatorica 26 (2006), 521–532.
P. Erdős: On some extremal problems in graph theory, Israel. J. Math. 3 (1965), 113–116.
J. Geelen, B. Gerards, B. Reed, P. Seymour and A. Vetta: On the odd-minor variant of Hadwiger’s conjecture, J. Combin. Theory Ser. B 99 (2009), 20–29.
H. Hadwiger: Über eine Klassifikation der Streckenkomplexe, Vierteljschr. Naturforsch. Ges. Zürich 88 (1943), 133–142.
T. R. Jensen and B. Toft: Graph coloring problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication.
K. Kawarabayashi: Note on coloring graphs without odd-Kk-minors, J. Combin. Theory Ser. B 99 (2009), 728–731.
A. V. Kostochka: The minimum Hadwiger number for graphs with a given mean degree of vertices, Metody Diskret. Analiz. 38 (1982), 37–58.
A. V. Kostochka: Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica 4 (1984), 307–316.
K.-i. Kawarabayashi and B. Reed: Highly parity linked graphs, Combinatorica 29 (2009), 215–225.
K.-i. Kawarabayashi and Z.-X. Song: Some remarks on the odd Hadwiger’s conjecture, Combinatorica 27 (2007), 429–438.
W. Mader: Existence of n-times connected subgraphs in graphs having large edge density, in: Essays from the Mathematical Seminar of the University of Hamburg, volume 37, 86–97, 1972.
S. Norin and Z.-X. Song: Breaking the degeneracy barrier for coloring graphs with no Kt minor, 2019. arXiv:1910.09378.
L. Postle: Halfway to Hadwiger’s Conjecture, 2019. arXiv:1911.01491.
P. Seymour: Hadwiger’s conjecture, in: Open problems in mathematics, 417–437, Springer, 2016.
A. Thomason: An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 261–265.
A. Thomason: The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001), 318–338.
R. Thomas and P. Wollan: An improved linear edge bound for graph linkages, European J. Combin. 26 (2005), 309–324.
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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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