Degeneracy and Colorings of Squares of Planar Graphs without 4-Cycles

Abstract

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if G is such a graph, then G2 is (Δ(G) + 72)-degenerate. This implies an upper bound of Δ(G) ∣ 73 on the chromatic number of G2 as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon-Tarsi number, and correspondence chromatic number. We also show that if Δ(G) is sufficiently large, then the upper bounds on each of these parameters of G2 can all be lowered to Δ(G) + 2 (which is best possible). To complement these results, we show that 4-cycles are unique in having this property. Specifically, let S be a finite list of positive integers, with 4 ∉ S. For each constant C, we construct a planar graph Gs,c with no cycle with length in S, but for which χ(G2S,C ) >Δ(Gs,c) + C.

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Acknowledgments

Thanks to four referees for their feedback; in particular, one wrote a very thorough and useful report. Thanks also to Zdenĕk Dvoĩák and Jean-Sébastien Sereni for their helpful comments after carefully reading Section 3. Zdenĕk caught a few errors in an earlier version of this paper.

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Correspondence to Daniel W. Cranston.

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Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07043049), and also by the Hankuk University of Foreign Studies Research Fund.

This research is partially supported by NSA Grant H98230-15-1-0013.

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Choi, I., Cranston, D.W. & Pierron, T. Degeneracy and Colorings of Squares of Planar Graphs without 4-Cycles. Combinatorica 40, 625–653 (2020). https://doi.org/10.1007/s00493-019-4014-3

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Mathematics Subject Classification (2010)

  • 05C15