Skip to main content
Springer Nature Link
Log in

Acyclic edge coloring of planar graphs without a \(3\)-cycle adjacent to a \(6\)-cycle

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

An acyclic edge coloring of a graph \(G\) is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index \(a'(G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an acyclic edge coloring using \(k\) colors. Fiamč ik (Math Slovaca 28:139–145, 1978) and later Alon et al. (J Graph Theory 37:157–167, 2001) conjectured that \(a'(G)\le \Delta +2\) for any simple graph \(G\) with maximum degree \(\Delta \). In this paper, we confirm this conjecture for planar graphs without a \(3\)-cycle adjacent to a \(6\)-cycle.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Basavaraju M, Chandran LS (2008) Acyclic edge coloring of subcubic graphs. Discret Math 308:6650–6653

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Molloy M, Reed B (1998) Further algorithmic aspects of Lov ász local lemma. in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 524–529.

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

  • Wang Y, Sheng P (2012) Acyclic edge coloring of sparse graphs. Discret Math 312:3561–3573

    Article  MATH  MathSciNet  Google Scholar 

  • Wang W, Shu Q, Wang Y (2012) Every \(4\)-regular graph is acyclically edge-\(6\)-colorable. Submitted, http://arxiv.org/abs/1209.2471v1

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

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

Download references

Acknowledgments

Supported partially by NSFC (No. 11301035) and Health Management and Health Economics Innovation Team, Beijing University of Chinese Medicine. Supported partially by NSFC (No. 11271006).

Author information

Authors and Affiliations

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

    Yiqiao Wang

  2. School of Science, Hangzhou Dianzhi University, Hangzhou , 310018, China

    Qiaojun Shu

  3. School of Mathematics and System Sciences, Shandong University, Jinan, 250100, China

    Jian-Liang Wu & Wenwen Zhang

Authors
  1. Yiqiao Wang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Qiaojun Shu
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Jian-Liang Wu
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Wenwen Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yiqiao Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Shu, Q., Wu, JL. et al. Acyclic edge coloring of planar graphs without a \(3\)-cycle adjacent to a \(6\)-cycle. J Comb Optim 28, 692–715 (2014). https://doi.org/10.1007/s10878-014-9765-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-014-9765-6

Keywords