Skip to main content

The r-dynamic chromatic number of planar graphs without 4-,5-cycles

Journal of Combinatorial Optimization volume 45, Article number: 47 (2023) Cite this article

Abstract

Let G be a graph and k, r be two positive integers. A (k, r)-dynamic coloring \(\varphi \) of G is a proper k-coloring such that \(\mid \varphi (N_{G}(v))\mid \ge \) \(\min \{d_{G}(v),r\}\) for each \(v\in V(G)\). In this paper, we prove that the r-dynamic chromatic number of planar graphs without 4-,5-cycles, and intersecting 3-cycles is at most \(r+3\) if \(r\ge 14\).

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Data availibility

Enquiries about data availability should be directed to the authors.

References

  • Borodin OV, Ivanova AO (2009) List 2-distance (\(\Delta + 2\))-coloring of planar graphs with girth six. Eur J Comb 30:1257–1262

    Article  MathSciNet  MATH  Google Scholar 

  • Bu Y, Cao J (2021) 2-Distance coloring of planar graph. Discrete Appl Math 13:2150007

    MathSciNet  MATH  Google Scholar 

  • Bu Y, Yang R, Zhu H (2022) The r-dynamic chromatic number of planar graphs without special short cycles. In: AAIM 2022, LNCS 13513

  • La H, Montassier M, Pinlou A et al (2021) r-Hued (r+1)-coloring of planar graphs with girth at least 8 for r\(\ge 9\). Eur J Comb 91:103219

    Article  MathSciNet  MATH  Google Scholar 

  • Montgomery B (2001) Dynamic coloring of graphs. West Virginia University, Morgantwon

    Book  Google Scholar 

  • Song H, Lai H (2018) Upper bounds of r-hued colorings of planar graphs. Discrete Appl Math 243:262–269

    Article  MathSciNet  MATH  Google Scholar 

  • Song H, Lai H, Wu J (2016) On r-hued coloring of planar graphs with girth at least 6. Discrete Appl Math 198:251–263

    Article  MathSciNet  MATH  Google Scholar 

  • Wang X (2020) r-Dynamic coloring of planar graphs. Zhejiang Noamal University, Jinhua

    Google Scholar 

  • Wegner G (1977) Graphs with given diameter and a coloring problem. Technical Report, University of Dortmund

  • Zhu J, Bu Y (2018) Minimum 2-distance coloring of planar graphs and channel assignment. J Comb Optim 36:55–64

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu H, Gu Y, Sheng J, Lv X (2018) List 2-distance (\(\Delta + 3\))-coloring of planar graphswithout 4,5-cycles. J Comb Optim 36:1411–1424

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was supported by National Science Foundation of China under Grant Nos. 11271334, 12201569, 11901243.

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Ruiying Yang, Yuehua Bu & Hongguo Zhu

  2. Department of Basics, Zhejiang Guangsha Vocational and Technical University of Construction, Dongyang, 322100, Zhejiang, China

    Yuehua Bu

  3. College of Data Science, Jiaxing University, Jiaxing, 314000, Zhejiang, China

    Junlei Zhu

Authors
  1. Ruiying Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuehua Bu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junlei Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongguo Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongguo Zhu.

Ethics declarations

Conflict of interest

We declare that we have no conflicts of interest to this work. We also declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by research Grants NSFC (11271334, 12201569, 11901243).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yang, R., Bu, Y., Zhu, J. et al. The r-dynamic chromatic number of planar graphs without 4-,5-cycles. J Comb Optim 45, 47 (2023). https://doi.org/10.1007/s10878-022-00985-5

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10878-022-00985-5

Keywords

  • r-Dynamic coloring
  • Planar graphs
  • Cycle
  • Discharging

Mathematics Subject Classification

  • 05C10
  • 05C15