Abstract
Let \(\mathrm{col_g}(G)\) be the game coloring number of a given graph G. Define the game coloring number of a family of graphs \(\mathcal {H}\) as \(\mathrm{col_g}(\mathcal {H}) := \max \{\mathrm{col_g}(G):G \in \mathcal {H}\}.\) Let \(\mathcal {P}_k\) be the family of planar graphs of girth at least k. We show that \(\mathrm{col_g}(\mathcal {P}_7) \le 5.\) This result extends a result about the game coloring number by Wang and Zhang [10] (\(\mathrm{col_g}(\mathcal {P}_8) \le 5).\) We also show that these bounds are sharp by constructing a graph G where \(G \in \mathcal {P}_k\) for each \(k \le 8\) such that \(\mathrm{col_g}(G)=5.\) As a consequence, \(\mathrm{col_g}(\mathcal {P}_k)=5\) for \(k =7,8.\)
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Acknowledgements
This work draws substantially on ideas from the paper by Kierstead [6] and the paper by Sekiguchi [9]. Without those ideas, this work would not be possible. We would like to express our gratitude to them for their papers. We would like to thank the referee for careful reading and valuable comments. The second author was supported by the Commission on Higher Education and the Thailand Research Fund under grant RSA5780014.
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Nakprasit, K.M., Nakprasit, K. The Game Coloring Number of Planar Graphs with a Specific Girth. Graphs and Combinatorics 34, 349–354 (2018). https://doi.org/10.1007/s00373-018-1877-9
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DOI: https://doi.org/10.1007/s00373-018-1877-9