Abstract
A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle (Comb Theory Ser B 129:38–54, 2018). Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring (Liu and Li, Discrete Math 342:623–627, 2019; Liu et al., Discrete Math 342(1):178–189, 2019; Kim and Ozeki, A note on a Brooks type theorem for DP-coloring, arXiv:1709.09807, 2019; Kim and Yu, Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, arXiv:1712.08999, 2019; Sittitrai and Nakprasit, Every planar graph without i-cycles adjacent simultaneously to j-cycles and k-cycles is DP-4-colorable when \(\{i,j,k\}=\{3,4,5\}\), arXiv:1801.06760, 2019; Yin and Yu, Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable, arXiv:1809.00925, 2019). We note that list-coloring results do not always extend to DP-coloring results, as shown in Bernshteyn and Kostochka (On differences between DP-coloring and list coloring, arXiv:1705.04883, 2019). Our main result in this paper is to prove that every planar graph without cycles of length \(\{4, a, b, 9\}\) for \(a, b \in \{6, 7, 8\}\) is DP-3-colorable, extending three existing results (Shen and Wang, Inf Process Lett 104:146–151, 2007; Wang and Shen, Discrete Appl Math 159:232–239, 2011; Whang et al., Inf Process Lett 105:206–211, 2008) on 3-choosability of planar graphs.
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References
Alon, N., Tarsi, M.: Colorings and orientations of graphs. Combinatorica 12, 125–134 (1992)
Bernshteyn, A., Kostochka, A.: On differences between DP-coloring and list coloring (2019). arXiv:1705.04883 (Preprint)
Dvořák, Z., Postle, L.: Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8. J. Comb. Theory Ser. B 129, 38–54 (2018)
Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs, Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium, vol. XXVI, pp. 125–157 (1979)
Liu, R., Li, X.: Every planar graph without 4-cycles adjacent to two triangles is DP-4-colorable. Discrete Math. 342, 623–627 (2019)
Liu, R., Loeb, S., Yin, Y., Yu, G.: DP-3-coloring of some planar graphs. Discrete Math. 342(1), 178–189 (2019)
Kim, S.-J., Ozeki, K.: A note on a Brooks type theorem for DP-coloring (2019). arXiv:1709.09807 (preprint)
Kim, S.-J., Yu, X.: Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable (2019). arXiv:1712.08999 (preprint)
Shen, L., Wang, Y.: A sufficient condition for a planar graph to be \(3\)-choosable. Inf. Process. Lett. 104, 146–151 (2007)
Sittitrai, P., Nakprasit, K.: Every planar graph without \(i\)-cycles adjacent simultaneously to \(j\)-cycles and \(k\)-cycles is DP-4-colorable when \(\{i,j,k\}=\{3,4,5\}\) (2019). arXiv:1801.06760 (preprint)
Thomassen, C.: Every planar graph is 5-choosable. J. Comb. Theory Ser. B 62, 180–181 (1994)
Thomassen, C.: 3-list-coloring planar graphs of girth 5. J. Comb. Theory Ser. B 64, 101–107 (1995)
Wang, Y., Shen, L.: Planar graphs without cycles of length \(4,7,8\) or \(9\) are \(3\)-choosable. Discrete Appl. Math. 159, 232–239 (2011)
Whang, Y., Lu, H., Chen, M.: A note on \(3\)-choosability of planar graphs. Inf. Process. Lett. 105, 206–211 (2008)
Vizing, V.G.: Vertex colorings with given colors (in Russian). Diskret. Anal. 29, 3–10 (1976)
Yin, Y., Yu, G.: Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable (2019). arXiv:1809.00925 (preprint)
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The research of the last author was supported in part by the Natural Science Foundation of China (11728102) and the NSA grant H98230-16-1-0316.
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Liu, R., Loeb, S., Rolek, M. et al. DP-3-Coloring of Planar Graphs Without 4, 9-Cycles and Cycles of Two Lengths from \(\{6,7,8\}\). Graphs and Combinatorics 35, 695–705 (2019). https://doi.org/10.1007/s00373-019-02025-2
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DOI: https://doi.org/10.1007/s00373-019-02025-2