Skip to main content
SpringerLink
Log in

Acyclic Edge Coloring of 4-Regular Graphs Without 3-Cycles

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

A proper edge coloring is called acyclic if no bichromatic cycles are produced. It was conjectured that every simple graph G with maximum degree \(\varDelta \) is acyclically edge-\((\varDelta +2)\)-colorable. Basavaraju and Chandran (J Graph Theory 61:192–209, 2009) confirmed the conjecture for non-regular graphs G with \(\varDelta =4\). In this paper, we extend this result by showing that every 4-regular graph G without 3-cycles is acyclically edge-6-colorable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Alon, N., Sudakov, B., Zaks, A.: Acyclic edge colorings of graphs. J. Graph Theory 37, 157–167 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersen, L.D., Máčajová, E., Mazák, J.: Optimal acyclic edge-coloring of cubic graphs. J. Graph Theory 71, 353–364 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Basavaraju, M., Chandran, L.S.: Acyclic edge coloring of subcubic graphs. Discrete Math. 308, 6650–6653 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Basavaraju, M., Chandran, L.S.: Acyclic edge coloring of graphs with maximum with degree 4. J. Graph Theory 61, 192–209 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Basavaraju, M., Chandran, L.S.: Acyclic edge coloring of 2-degenerate graphs. J. Graph Theory 69, 1–27 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Basavaraju, M., Chandran, L.S., Cohen, N., Havet, F., Müller, T.: Acyclic edge-coloring of planar graphs. SIAM J. Discrete Math. 25, 463–478 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dong, W., Xu, B.: Some results on acyclic edge coloring of plane graphs. Inf. Process. Lett. 110, 887–892 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Esperet, L., Parreau, A.: Acyclic edge-coloring using entropy compression. Eur. J. Comb. 34, 1019–1027 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fiamčik, J.: The acyclic chromatic class of a graph. Math. Slovaca 28, 139–145 (1978) (in Russian)

  11. Fiedorowicz, A., Haluszczak, M., Narayanan, N.: About acyclic edge colourings of planar graphs. Inf. Process. Lett. 108, 412–417 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hou, J., Wu, J., Liu, G., Liu, B.: Acyclic edge chromatic number of outerplanar graphs. J. Graph Theory 64, 22–36 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Molloy, M., Reed, B.: Further algorithmic aspects of Lovász local lemma. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pp. 524–529 (1998)

  14. Ndreca, S., Procacci, A., Scoppola, B.: Improved bounds on coloring of graphs. Eur. J. Comb. 33, 592–609 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shu, Q., Wang, W.: Acyclic chromatic indices of planar graphs with girth at least five. J. Comb. Optim. 23, 140–157 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shu, Q., Wang, W., Wang, Y.: Acyclic chromatic indices of planar graphs with girth at least four. J. Graph Theory 73, 386–399 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shu, Q., Wang, W., Wang, Y.: Acyclic edge coloring of planar graphs without 5-cycles. Discrete Appl. Math. 160, 1211–1223 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Skulrattanakulchai, S.: Acyclic colorings of subcubic graphs. Inf. Process. Lett. 92, 161–167 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vizing, V.: On an estimate of the chromatic index of a \(p\)-graph. Diskret Analiz 3, 25–30 (1964)

    Google Scholar 

  20. Wang, W., Shu, Q.: Acyclic chromatic indices of \(K_{4}\)-minor free graphs. Sci. China Ser. A 41, 733–744 (2011)

    Google Scholar 

  21. Wang, W., Shu, Q., Kan, W., Wang, P.: Acyclic chromatic indices of planar graphs with large girth. Discrete Appl. Math. 159, 1239–1253 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, W., Shu, Q., Wang, Y.: Acyclic edge coloring of planar graphs without 4-cycles. J. Comb. Optim. 25, 562–586 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, W., Shu, Q., Wang, Y.: A new upper bound on the acyclic chromatic indices of planar graphs. Eur. J. Comb. 34, 338–354 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, Y., Shu, Q., Wu, J., Zhang, W.: Acyclic edge coloring of planar graphs without 3-cycles adjacent to 6-cycles. J. Comb. Optim. 28, 692–715 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, T., Zhang, Y.: Further result on acyclic chromatic index of planar graphs. Discrete Appl. Math. 201, 228–247 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Qiaojun Shu

  2. School of Management, Beijing University of Chinese Medicine, Beijing, 100029, China

    Yiqiao Wang

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

    Yulai Ma & Weifan Wang

Authors
  1. Qiaojun Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiqiao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yulai Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Weifan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weifan Wang.

Additional information

Qiaojun Shu research was supported partially by ZJNSF (LQ15A010010) and NSFC (No. 11601111). Yiqiao Wang research was supported partially by NSFC (Nos. 11301035 and 11671053); Weifan Wang research was supported partially by NSFC (Nos. 11071223 and 11371328).

Communicated by Xueliang Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shu, Q., Wang, Y., Ma, Y. et al. Acyclic Edge Coloring of 4-Regular Graphs Without 3-Cycles. Bull. Malays. Math. Sci. Soc. 42, 285–296 (2019). https://doi.org/10.1007/s40840-017-0484-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-017-0484-x

Keywords

Mathematics Subject Classification

Search

Navigation