Abstract
A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at distance at most 2 from each other receive different colors. Suppose that G is a planar graph with girth 5 and maximum degree \(\Delta \). We prove that G admits a 2-distance \(\Delta +7\) coloring, which improves the result of Dong and Lin (J Comb Optim 32(2):645–655, 2016). Moreover, we prove that G admits a 2-distance \(\Delta +6\) coloring when \(\Delta \ge 10\).
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References
Yuehua B, Shang C (2016) List 2-distance coloring of planar graphs without short cycles. Discrete Math Algorithms Appl 8(01):1650013
Bu Y, Zhu X (2012) An optimal square coloring of planar graphs. J Comb Optim 24(4):580–592
Dong W, Lin W (2016) An improved bound on 2-distance coloring plane graphs with girth 5. J Comb Optim 32:645–655
Dong W, Lin W (2017) On 2-distance coloring of plane graphs with girth 5. Discret Appl Math 217:495–505
Hartke Stephen G, Jahanbekam S, Thomas B (2016) The chromatic number of the square of subcubic planar graphs. arXiv preprint arXiv:1604.06504
La H (2022) 2-distance list (\(\delta \)+ 3)-coloring of sparse graphs. Graphs Comb 38(6):167
La H, Montassier M (2022) 2-distance list (\(\delta \)+ 2)-coloring of planar graphs with girth at least 10. J Comb Optim 44(2):1356–1375
Molloy M, Salavatipour MR (2005) A bound on the chromatic number of the square of a planar graph. J Comb Theory Ser B 94:189–213
Thomassen C (2018) The square of a planar cubic graph is 7-colorable. J Comb Theory Ser B 128:192–218
van den Heuvel J, McGuinness S (2003) Coloring the square of a planar graph. J Graph Theory 42(2):110–124
Wegner G (1977) Graphs with given diameter and a coloring problem. Technical report, University of Dormund
West DB (2001) Introduction to graph theory, vol 2. Prentice hall, Upper Saddle River
Acknowledgements
This research was supported by TUBITAK (The Scientific and Technological Research Council of Türkiye) under the project number 122F250. We thank anonymous referees for carefully reading and for their invaluable comments and suggestions which have improved the presentation of this paper.
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This research was supported by TUBITAK (Grant Number 122F250).
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Deniz, Z. Some results on 2-distance coloring of planar graphs with girth five. J Comb Optim 47, 68 (2024). https://doi.org/10.1007/s10878-024-01169-z
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DOI: https://doi.org/10.1007/s10878-024-01169-z