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Packing Chromatic Number of Subdivisions of Cubic Graphs

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Abstract

A packing k-coloring of a graph G is a partition of V(G) into sets \(V_1,\ldots ,V_k\) such that for each \(1\le i\le k\) the distance between any two distinct \(x,y\in V_i\) is at least \(i+1\). The packing chromatic number, \(\chi _p(G)\), of a graph G is the minimum k such that G has a packing k-coloring. For a graph G, let D(G) denote the graph obtained from G by subdividing every edge. The questions on the value of the maximum of \(\chi _p(G)\) and of \(\chi _p(D(G))\) over the class of subcubic graphs G appear in several papers. Gastineau and Togni asked whether \(\chi _p(D(G))\le 5\) for any subcubic G, and later Brešar, Klavžar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that \(\chi _p(G)\) is not bounded in the class of subcubic graphs G. In contrast, in this paper we show that \(\chi _p(D(G))\) is bounded in this class, and does not exceed 8.

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References

  1. Argiroffo, G., Nasini, G., Torres, P.: The packing coloring problem for lobsters and partner limited graphs. Discrete Appl. Math. 164, 373–382 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balogh, J., Kostochka, A., Liu, X.: Packing chromatic number of cubic graphs. Discrete Math. 341, 474–483 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brešar, B., Ferme, J.: An infinite family of subcubic graphs with unbounded packing chromatic number. Discrete Math. 341, 2337–2342 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brešar, B., Klavžar, S., Rall, D.F.: On the packing chromatic number of Cartesian products, hexagonal lattice, and trees. Discrete Appl. Math. 155, 2302–2311 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Brešar, B., Klavžar, S., Rall, D.F.: Packing chromatic number of base-3 Sierpiński graphs. Graphs Combin. 32, 1313–1327 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brešar, B., Klavžar, S., Rall, D.F., Wash, K.: Packing chromatic number under local changes in a graph. Discrete Math. 340, 1110–1115 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brešar, B., Klavžar, S., Rall, D.F., Wash, K.: Packing chromatic number, (1,1,2,2)-colorings, and characterizing the Petersen graph. Aequationes Math. 91, 169–184 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brešar, B., Klavžar, S., Rall, D.F., Wash, K.: Packing chromatic number versus chromatic and clique number. Aequationes Math. 92, 497–513 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Czap, J., Jendrol’, S.: Facial packing edge-coloring of plane graphs. Discrete Appl. Math. 213, 71–75 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fiala, J., Golovach, P.A.: Complexity of the packing coloring problem for trees. Discrete Appl. Math. 158, 771–778 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fiala, J., Klavžar, S., Lidický, B.: The packing chromatic number of infinite product graphs. Eur. J. Combin. 30, 1101–1113 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gastineau, N.: Dichotomies properties on computational complexity of S-packing coloring problems. Discrete Math. 338, 1029–1041 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gastineau, N., Togni, O.: S-packing colorings of cubic graphs. Discrete Math. 339, 2461–2470 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gastineau, N., Holub, P., Togni, O.: On packing chromatic number of subcubic outerplanar graphs. arXiv:1703.05023

  15. Gastineau, N., Kheddouci, H., Togni, O.: Subdivision into \(i\)-packing and \(S\)-packing chromatic number of some lattices. ARS Math. Contemp. 9, 331–354 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goddard, W., Xu, H.: The S-packing chromatic number of a graph. Discuss. Math. Graph Theory 32, 795–806 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goddard, W., Xu, H.: A note on S-packing colorings of lattices. Discrete Appl. Math. 166, 255–262 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Goddard, W., Hedetniemi, S.M., Hedetniemi, S.T., Harris, J.M., Rall, D.F.: Broadcast chromatic numbers of graphs. ARS Combin. 86, 33–49 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Korže, D., Vesel, A.: On the packing chromatic number of square and hexagonal lattice. ARS Math. Contemp. 7, 13–22 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Laiche, D., Bouchemakh, I., Sopena, É.: On the packing coloring of undirected and oriented generalized theta graphs. Australas. J. Combin. 66, 310–329 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Laiche, D., Bouchemakh, I., Sopena, É.: Packing coloring of some undirected and oriented coronae graphs. Discuss. Math. Graph Theory 37, 665–690 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sloper, C.: An eccentric coloring of trees. Australas. J. Combin. 29, 309–321 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Torres, P., Valencia-Pabon, M.: The packing chromatic number of hypercubes. Discrete Appl. Math. 190–191, 127–140 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Sandi Klavžar, Douglas West, and the referees for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, USA

    József Balogh, Alexandr Kostochka & Xujun Liu

  2. Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russian Federation

    József Balogh

  3. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    Alexandr Kostochka

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  1. József Balogh
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Correspondence to Xujun Liu.

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Research of József Balogh is partially supported by NSF Grant DMS-1500121 and by the Langan Scholar Fund (UIUC). Research of Alexandr Kostochka is supported in part by NSF Grant DMS-1600592 and Grants 18-01-00353A and 16-01-00499 of the Russian Foundation for Basic Research.

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Balogh, J., Kostochka, A. & Liu, X. Packing Chromatic Number of Subdivisions of Cubic Graphs. Graphs and Combinatorics 35, 513–537 (2019). https://doi.org/10.1007/s00373-019-02016-3

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  • DOI: https://doi.org/10.1007/s00373-019-02016-3

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