Graphs and Combinatorics

, Volume 35, Issue 3, pp 695–705 | Cite as

DP-3-Coloring of Planar Graphs Without 4, 9-Cycles and Cycles of Two Lengths from \(\{6,7,8\}\)

  • Runrun Liu
  • Sarah Loeb
  • Martin Rolek
  • Yuxue Yin
  • Gexin YuEmail author
Original Paper


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.


DP-3-coloring Planar graphs Discharging List coloring 



  1. 1.
    Alon, N., Tarsi, M.: Colorings and orientations of graphs. Combinatorica 12, 125–134 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernshteyn, A., Kostochka, A.: On differences between DP-coloring and list coloring (2019). arXiv:1705.04883 (Preprint)
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Liu, R., Li, X.: Every planar graph without 4-cycles adjacent to two triangles is DP-4-colorable. Discrete Math. 342, 623–627 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Liu, R., Loeb, S., Yin, Y., Yu, G.: DP-3-coloring of some planar graphs. Discrete Math. 342(1), 178–189 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kim, S.-J., Ozeki, K.: A note on a Brooks type theorem for DP-coloring (2019). arXiv:1709.09807 (preprint)
  8. 8.
    Kim, S.-J., Yu, X.: Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable (2019). arXiv:1712.08999 (preprint)
  9. 9.
    Shen, L., Wang, Y.: A sufficient condition for a planar graph to be \(3\)-choosable. Inf. Process. Lett. 104, 146–151 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)
  11. 11.
    Thomassen, C.: Every planar graph is 5-choosable. J. Comb. Theory Ser. B 62, 180–181 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Thomassen, C.: 3-list-coloring planar graphs of girth 5. J. Comb. Theory Ser. B 64, 101–107 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Whang, Y., Lu, H., Chen, M.: A note on \(3\)-choosability of planar graphs. Inf. Process. Lett. 105, 206–211 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Vizing, V.G.: Vertex colorings with given colors (in Russian). Diskret. Anal. 29, 3–10 (1976)Google Scholar
  16. 16.
    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)

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCentral China Normal UniversityWuhanChina
  2. 2.Department of MathematicsThe College of William and MaryWilliamsburgUSA
  3. 3.Hampden-Sydney CollegeHampden-SdyneyUSA

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