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


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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  1. [1]

    N. Álon and M. Tarsi: Colorings and orientations of graphs, Combinatorial12 (1992), 125–134.

    MathSciNet  Article  Google Scholar 

  2. [2]

    O. Amini, L. Esperet and J. van den Heuvel: A unified approach to distance-two colouring of graphs on surfaces, Combinatorial33 (2013), 253–296.

    MathSciNet  Article  Google Scholar 

  3. [3]

    M. Bonamy, D. W. Cranston and L. Postle: Planar graphs of girth at least five are square (Δ + 2)-choosable, J. Combin. Theory Ser. B134 (2019), 218–238.

    MathSciNet  Article  Google Scholar 

  4. [4]

    O. V. Borodin, A. N. Glebov, A. O. Ivanova, T. K. Neustroeva and V. A. Tashkinov: Sufficient conditions for planar graphs to be 2-distance (Δ + 1)-colorable, Sib. Elektron. Mat. Izv.1 (2004), 129–141.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    D. W. Cranston and B. Jaeger: List-coloring the squares of planar graphs without 4-cycles and 5-cycles, J. Graph Theory85 (2017), 721–737.

    MathSciNet  Article  Google Scholar 

  6. [6]

    D. W. Cranston and D. B. West: An introduction to the discharging method via graph coloring, Discrete Math.340 (2017), 766–793.

    MathSciNet  Article  Google Scholar 

  7. [7]

    W. Dong and B. Xu: 2-distance coloring of planar graphs without 4-cycles and 5-cycles, SIAM J. Discrete Math., 33 (2019), 1297–1312.

    MathSciNet  Article  Google Scholar 

  8. [8]

    Z. Dvořák, D. Král, P. Nejedlý and R. Škrekovski: Coloring squares of planar graphs with girth six, European J. Combin.29 (2008), 838–849.

    MathSciNet  Article  Google Scholar 

  9. [9]

    F. Havet, J. van den Heuvel, C. McDiarmid and B. Reed: List colouring squares of planar graphs, July 2008, preprint available at

  10. [10]

    T. Kimball Jonas: Graph coloring analogues with a condition at distance two: L(2, 1)-labellings and list lambda-labellings, ProQuest LLC, Ann Arbor, MI, 1993, Thesis (Ph.D.)-University of South Carolina.

    Google Scholar 

  11. [11]

    M. Molloy and M. R. Salavatipour: A bound on the chromatic number of the square of a planar graph, J. Combin. Theory Ser. B94 (2005), 189–213.

    MathSciNet  Article  Google Scholar 

  12. [12]

    U. Schauz: Flexible color lists in Alon and Tarsi’s theorem, and time scheduling with unreliable participants, Electronic J. Combin.17 (2010), 13.

    MathSciNet  Article  Google Scholar 

  13. [13]

    W.-F. Wang and K.-W. Lih: Labeling planar graphs with conditions on girth and distance two, SIAM J. Discrete Math.17 (2003), 264–275.

    MathSciNet  Article  Google Scholar 

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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).

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Mathematics Subject Classification (2010)

  • 05C15