Abstract
We show that for any n ≥ 13, there exist graphs with chromatic number larger than n whose product has chromatic number n. Our construction is an adaptation of the construction of counterexamples to Hedetniemi’s conjecture devised by Shitov, and adapted by Zhu to graphs with relatively small chromatic numbers. The new tools we introduce are graphs with minimal colourings that are “wide” in the sense of Simonyi and Tardos, and generalised Mycielskians to settle the case n = 13.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Baum and M. Stiebitz: Coloring of graphs without short odd paths between vertices of the same color class (2005). Manuscript, 23 PP.
S. A. Burr, P. Erdős and L. Lovász: On graphs of Ramsey type, Ars Combinatoria 1 (1976), 167–190.
M. El-Zahar and N. Sauer: The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica 5 (1985), 121–126.
A. Gujgiczer and G. Simonyi: On multichromatic numbers of widely colorable graphs, manuscript (2021), arXiv:2102.03120.
A. Gyárfás, T. Jensen and M. Stiebitz: On graphs with strongly independent colorclasses, J. Graph. Theory 46 (2004), 1–14.
G. Hahn and C. Tardif: Graph homomorphisms: structure and symmetry, Graph symmetry (Montreal, PQ, 1996), 107–166, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences 497, Kluwer Academic Publishers, Dordrecht, 1997.
H. Hajiabolhassan: On colorings of graph powers, Discrete Math. 309 (2009), 4299–4305.
X. He and Y. Wigderson: Hedetniemi’s conjecture is asympototically false, J. Combin. Theory Ser. B 146 (2021), 485–494.
S. Hedetniemi: Homomorphisms of graphs and automata, Technical Report 03105-44-T, University of Michigan, 1966.
L. Lovász: Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory Ser. A 25 (1978), 319–324.
Y. Shitov: Counterexamples to Hedetniemi’s Conjecture, Ann. of Math. (2) 190 (2019), 663–667.
G. Simonyi and G. Tardos: Local chromatic number, Ky Fan’s Theorem and circular colorings, Combinatorica 26 (2006), 587–626.
C. Tardif and X. Zhu: A note on Hedetniemi’s conjecture, Stahl’s conjecture and the Poljak-Rödl function, Electron. J. Combin. 26 (2019), #P4.32.
M. Wrochna: On inverse powers of graphs and topological implications of Hedetniemi’s conjecture, J. Combin. Theory Ser. B 139 (2019), 267–295.
M. Wrochna: Smaller counterexamples to Hedetniemi’s conjecture, manuscript (2020), arXiv:2012.13558.
X. Zhu: A note on the Poljak—Rödl function, Electron. J. Combin. 27 (2020), #P3.2.
X. Zhu: Relatively small counterexamples to Hedetniemi’s conjecture, J. Combin. Theory Ser. B 146 (2021), 141–150.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tardif, C. The Chromatic Number of the Product of 14-Chromatic Graphs Can BE 13. Combinatorica 42, 301–308 (2022). https://doi.org/10.1007/s00493-021-4781-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-021-4781-5