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Edge Colorings of the Direct Product of Two Graphs

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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.

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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 

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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.

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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
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  2. Davide Mazza
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  3. Norma Zagaglia Salvi
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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).

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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

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

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