Abstract
Given a graph G and a list assignment L(v) for each vertex of v of G, a proper L-list-coloring of G is a function that maps every vertex to a color in L(v) such that no pair of adjacent vertices have the same color. We say that a graph is k-list-colorable when every vertex v has a list of colors of size at least k. A 2-distance coloring is a coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance list (\(\Delta +2\))-coloring for planar graphs with girth at least 10 and maximum degree \(\Delta \ge 4\).
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References
Bonamy M, Cranston D, Postle L (2019) Planar graphs of girth at least five are square (\(\Delta +2\))-choosable. J of Comb Theory, Ser B 134:218–238
Bonamy M, Lévêque B, Pinlou A (2014) 2-distance coloring of sparse graphs. J of Graph Theory 77(3):190–218
Bonamy M, Lévêque B, Pinlou A (2014) Graphs with maximum degree \(\Delta \ge 17\) and maximum average degree less than 3 are list 2-distance (\(\Delta + 2\))-colorable. Discrete Math 317:19–32
Borodin OV, Glebov AN, Ivanova AO, Neutroeva TK, Tashkinov VA (2004) Sufficient conditions for the 2-distance (\(\Delta +1\))-colorability of plane graphs. Sibirskie Elektronnye Matematicheskie Izvestiya 1:129–141
Borodin OV, Ivanova AO (2012) List 2-facial 5-colorability of plane graphs with girth at least 12. Discrete Math 312:306–314
Bu Y, Lv X, Yan X (2015) The list 2-distance coloring of a graph with \(\Delta (G)=5\). Discrete Math, Algorithms and Appl 7(2):1550017
Bu Y, Shang C (2016) List 2-distance coloring of planar graphs without short cycles. Discrete Math, Algorithms and Appl 8(1):1650013
Bu Y, Zhu J (2018) Channel Assignment with r-Dynamic Coloring: 12th International Conference, AAIM 2018, Dallas, TX, USA, December 3–4, 2018, Proceedings pp 36–48
Bu Y, Zhu X (2012) An optimal square coloring of planar graphs. J of Comb Optim 24:580–592
Cranston D, Erman R, Škrekovski R (2014) Choosability of the square of a planar graph with maximum degree four. Australian J of Comb 59(1):86–97
Cranston D, Kim S-J (2008) List-coloring the square of a subcubic graph. J of Graph Theory 1:65–87
Dong W, Lin W (2016) An improved bound on 2-distance coloring plane graphs with girth 5. J of Comb Optim 32(2):645–655
Dong W, Lin W (2017) On 2-distance coloring of plane graphs with girth 5. Discrete Appl Math 217:495–505
Dong W, Xu B (2017) 2-distance coloring of planar graphs with girth 5. J of Comb Optim 34:1302–1322
Dvořák Z, Kràl D, Nejedlỳ P, Škrekovski R (2008) Coloring squares of planar graphs with girth six. European J of Comb 29(4):838–849
Hartke SG, Jahanbekam S, Thomas B (2018) The chromatic number of the square of subcubic planar graphs. arXiv:1604.06504
Havet F, Van Den Heuvel J, McDiarmid C, Reed B (2017) List colouring squares of planar graphs. arXiv:0807.3233
Ivanova AO (2011) List 2-distance (\(\Delta \)+1)-coloring of planar graphs with girth at least 7. J of Appl and Industrial Math 5(2):221–230
Kramer F, Kramer H (1969) Ein Färbungsproblem der Knotenpunkte eines Graphen bezüglich der Distanz p. Revue Roumaine de Mathématiques Pures et Appliquées 14(2):1031–1038
Kramer F, Kramer H (1969) Un problème de coloration des sommets d’un graphe. Comptes Rendus Mathématique Académie des Sci, Paris 268:46–48
La H (2021) 2-distance list \((\Delta +3)\)-coloring of sparse graphs. arXiv:2105.01684
La H, Montassier M (2021a) 2-distance 4-coloring of planar subcubic graphs with girth at least 21. arXiv:2106.03587
La H, Montassier M (2021b) 2-distance \((\Delta +1)\)-coloring of sparse graphs using the potential method. arXiv:2103.11687
La H, Montassier M (2021c) 2-distance \((\Delta +2)\)-coloring of sparse graphs. arXiv:2109.11927
La H, Montassier M, Pinlou A, Valicov P (2021) \(r\)-hued \((r+1)\)-coloring of planar graphs with girth at least 8 for \(r\ge 9\). European J of Comb 91:103219
Lih K-W, Wang W-F, Zhu X (2003) Coloring the square of a \(K_4\)-minor free graph. Discrete Math 269(1):303–309
Thomassen C (2018) The square of a planar cubic graph is 7-colorable. J of Comb Theory, Ser B 128:192–218
Wegner G (1977) Graphs with given diameter and a coloring problem. Technical report, University of Dormund
Funding
This work was partially supported by the grant HOSIGRA funded by the French National Research Agency (ANR, Agence Nationale de la Recherche) under the contract number ANR-17-CE40-0022.
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La, H., Montassier, M. 2-Distance list \((\Delta +2)\)-coloring of planar graphs with girth at least 10. J Comb Optim 44, 1356–1375 (2022). https://doi.org/10.1007/s10878-022-00883-w
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DOI: https://doi.org/10.1007/s10878-022-00883-w