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Acyclic Choosability of Graphs with Bounded Degree

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Abstract

An acyclic colouring of a graph G is a proper vertex colouring such that every cycle uses at least three colours. For a list assignment L = {L(v)∼ v ∈ V(G)}, if there exists an acyclic colouring ρ such that ρ(v) ∈ L(v) for each v ∈ V(G), then ρ is called an acyclic L-list colouring of G. If there exists an acyclic L-list colouring of G for any L with ∣L(v)∣> k for each v ∈ V (G), then G is called acyclically k-choosable. In this paper, we prove that every graph with maximum degree Δ ≤ 7 is acyclically 13-choosable. This upper bound is first proposed. We also make a more compact proof of the result that every graph with maximum degree Δ ≤ 3 (resp., Δ ≤ 4) is acyclically 4-choosable (resp., 5-choosable).

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References

  1. Alon, N., Mcdiarmid, C., Reed, B.: Acyclic coloring of graphs. Random Struct. Algor., 2, 277–288 (1991)

    Article  MathSciNet  Google Scholar 

  2. Bollobás, B.: Modern Graph Theory, Springer-Verlag, New York, 1998

    Book  Google Scholar 

  3. Borodin, O. V., Ivanova, A. O.: Acyclic 5-choosability of planar graphs without 4-cycles. Siberian Math. J., 52, 411–425 (2011)

    Article  MathSciNet  Google Scholar 

  4. Borodin, O. V., Ivanova, A. O.: Acyclic 4-choosability of planar graphs with no 4- and 5-cycles. J. Graph Theory, 72, 374–397 (2013)

    Article  MathSciNet  Google Scholar 

  5. Borodin, O. V., Fon-Der Flaass, D. G., Kostochka, A. V., et al.: Acyclic list 7-coloring of planar graphs. J. Graph Theory, 40, 83–90 (2002)

    Article  MathSciNet  Google Scholar 

  6. Borodin, O. V.: On acyclic colorings of planar graphs. Discrete Math., 25, 211–236 (1979)

    Article  MathSciNet  Google Scholar 

  7. Burstein M. I.: Every 4-valent graph has an acyclic 5-coloring (in Russian). Soobšč. Akad. Nauk Gruzin SSR, 93, 21–24 (1979)

    MATH  Google Scholar 

  8. Chen, M., Raspaud, A.: A sufficient condition for planar graphs to be acyclically 5-choosable. J. Graph Theory, 70, 135–151 (2012)

    Article  MathSciNet  Google Scholar 

  9. Dvořák, Z.: 3-choosability of planar graphs with (≤ 4)-cycles far apart. J. Combin. Theory, Ser. B, 104, 28–59 (2014)

    Article  MathSciNet  Google Scholar 

  10. Fertin, G., Raspaud, A.: Acyclic coloring of graphs of maximum degree Δ. European Conference on Combinatorics, Graph Theory and Applications, Berlin, Germany, 2005, 389–396

  11. Fertin, G., Raspaud, A.: Acyclic coloring of graphs of maximum degree five: nine colors are enough. Inform. Process. Lett., 105, 65–72 (2008)

    Article  MathSciNet  Google Scholar 

  12. Fiedorowicz, A., Sidorowicz, E.: Acyclic colorings of graphs with bounded degree. Sci. China Math., 59, 1427–1440 (2016)

    Article  MathSciNet  Google Scholar 

  13. Gonçalves, D., Montassier, M.: Acyclic choosability of graphs with small maximum degree. In: Graphtheoretic Concepts in Computer Science, Lecture Notes in Comput. Sci., Vol. 3787, Springer, Berlin, 2005, 239–248

    Chapter  Google Scholar 

  14. Grünbaum, B.: Acyclic colorings of planar graphs. Israel J. Math., 14, 390–408 (1973)

    Article  MathSciNet  Google Scholar 

  15. Hocquard, H.: Graphs with maximum degree 6 are acyclically 11-colorable. Inform. Process. Lett., 111, 748–753 (2011)

    Article  MathSciNet  Google Scholar 

  16. Hu, D., Huang, D., Wang, W., et al.: Planar graphs without chordal 6-cycles are 4-choosable. Discrete Appl. Math., 244, 116–123 (2018)

    Article  MathSciNet  Google Scholar 

  17. Kaul, H., Mudrock, J. A.: On the Alon—Tarsi number and chromatic-choosability of Cartesian products of graphs. Electron. J. Comb., 26, P1.3, 13 pp. (2019)

    Article  MathSciNet  Google Scholar 

  18. Kostochka, A. V., Stocker, C.: Graphs with maximum degree 5 are acyclically 7-colorable. Ars Math. Contemp., 4, 53–164 (2011)

    Article  MathSciNet  Google Scholar 

  19. Mohar, B.: Acyclic colorings of locally planar graphs. Eur. J. Combin., 26, 491–503 (2005)

    Article  MathSciNet  Google Scholar 

  20. Montassier, M., Ochem, P., Raspaud, A.: On the acyclic choosability of graphs. J. Graph Theory, 51, 281–300 (2006)

    Article  MathSciNet  Google Scholar 

  21. Wang, J., Miao, L.: Acyclic coloring of graphs with maximum degree at most six. Discrete Math., 342, 3025–3033 (2019)

    Article  MathSciNet  Google Scholar 

  22. Wang, J., Miao, L., Song, W.: Acyclic L-coloring of graphs with maximum degrees 5 and 6. Discrete Math., 343, 111584, 6 pp. (2020)

    Article  MathSciNet  Google Scholar 

  23. Wang, J., Miao, L., Song, W., et al.: Acyclic coloring of graphs with maximum degree 7. Graphs Combin., 37, 455–469 (2021)

    Article  MathSciNet  Google Scholar 

  24. Zhao, Y., Miao, L., Pang, S., et al.: Acyclic vertex coloring of graphs of maximum degree six. Discrete Math., 325, 17–22 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the editors and anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Management, Qufu Normal University, Rizhao, 276826, P. R. China

    Juan Wang

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, P. R. China

    Juan Wang, Lian Ying Miao & Jin Bo Li

  3. College of Engineering, Qufu Normal University, Rizhao, 276826, P. R. China

    Yun Long Liu

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  1. Juan Wang
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  2. Lian Ying Miao
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Correspondence to Juan Wang.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 11771443 and 11601510) and Shandong Province Natural Science Foundation (Grant No. ZR2017QF011)

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Wang, J., Miao, L.Y., Li, J.B. et al. Acyclic Choosability of Graphs with Bounded Degree. Acta. Math. Sin.-English Ser. 38, 560–570 (2022). https://doi.org/10.1007/s10114-022-0097-7

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  • DOI: https://doi.org/10.1007/s10114-022-0097-7

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