Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Edge Colorings of the Direct Product of Two Graphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

It is conjectured Zhang et al. (Appl Math Lett 15: 623–626, 2002) that if \(G\notin \{K_2,C_5\}\) is a connected graph, then there is a proper edge coloring of \(G\) using at most \(\Delta (G)+2\) colors that distinguishes vertices of \(G\) by means of their (naturally defined) color sets. In the paper several results concerning edge colorings of the direct product of two graphs are obtained that support the conjecture.

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

Similar content being viewed by others

References

  1. Balister, P.N., Győri, E., Lehel, J., Schelp, R.H.: Adjacent vertex distinguishing edge-colorings. SIAM J. Discrete Math. 21, 237–250 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baril, J.-L., Kheddouci, H., Togni, O.: Adjacent vertex distinguishing edge-colorings of meshes. Australas. J. Comb. 35, 89–102 (2006)

    MATH  MathSciNet  Google Scholar 

  3. Baril, J.-L., Kheddouci, H., Togni, O.: Vertex distinguishing edge- and total-colorings of Cartesian and other product graphs. Ars Comb. 107, 109–127 (2012)

    MATH  MathSciNet  Google Scholar 

  4. Edwards, K., Hor\(\check{\text{ n }}\)ák, M., Woźniak, M.: On the neighbour-distinguishing index of a graph. Graphs Comb. 22, 341–350 (2006)

  5. Frigerio, L., Lastaria, F.: Zagaglia Salvi, N.: Adjacent vertex distinguishing edge colorings of the direct product of a regular graph by a path or a cycle. Discuss. Math. Graph Theory 31, 547–557 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hatami, H.: \(\Delta +300\) is a bound on the adjacent vertex distinguishing edge chromatic number. J. Comb. Theory Ser. B 95, 246–256 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Holyer, I.: The NP-completeness of edge-colouring. SIAM J. Comput. 10, 718–720 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  8. Horňák, M., Huang, D., Wang, W.: On neighbour-distinguishing index of planar graphs, J. Graph Theory. doi:10.1002/jgt.21764.

  9. Imrich, W., Klavžar, S.: Product Graphs: Structure and Recognition. Wiley-Interscience, New York (2000)

    Google Scholar 

  10. Munarini, E.: Perelli Cippo, C., Zagaglia Salvi, N.: On the adjacent vertex distinguishing edge colorings of direct product of graphs. Quad. Mat. 28, 365–390 (2013)

    Google Scholar 

  11. Wang, W., Wang, Y.: Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree. J. Comb. Optim. 19, 471–485 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zhang, Z., Liu, L., Wang, J.: Adjacent strong edge coloring of graphs. Appl. Math. Lett. 15, 623–626 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank to anonymous referees for their remarks which helped to improve the presentation of results of the paper. They are especially grateful for the simple proof of Theorem 1.

Author information

Authors and Affiliations

  1. Institute of Mathematics, P.J. Šafárik University, Jesenná 5, 04001 , Košice, Slovakia

    Mirko Horňák

  2. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 , Milan, Italy

    Davide Mazza & Norma Zagaglia Salvi

Authors
  1. Mirko Horňák
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Davide Mazza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Norma Zagaglia Salvi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mirko Horňák.

Additional information

The work of the first author was supported by Science and Technology Assistance Agency under the contract No. APVV-0023-10 and by Grant VEGA 1/0652/12. The work of the remaining two authors was partially supported by MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Horňák, M., Mazza, D. & Zagaglia Salvi, N. Edge Colorings of the Direct Product of Two Graphs. Graphs and Combinatorics 31, 975–992 (2015). https://doi.org/10.1007/s00373-014-1413-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-014-1413-5

Keywords

Mathematics Subject Classfication (2010)

Search

Navigation