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Packing coloring of Sierpiński-type graphs

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Abstract

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in \{1,\ldots ,k\}\), where each \(V_i\) is an i-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs \(ST^n_3\) is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs \(S^n_G\) with respect to all connected graphs G of order 4.

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Acknowledgements

We wish to thank an anonymous reviewer for carefully checking the paper and suggesting a number of improvements for the presentation. B.B. acknowledges the financial support from the Slovenian Research Agency (Research Core Funding No. P1-0297 and the Project Grant J1-7110).

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Boštjan Brešar & Jasmina Ferme

  2. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Boštjan Brešar

  3. Faculty of Education, University of Maribor, Maribor, Slovenia

    Jasmina Ferme

Authors
  1. Boštjan Brešar
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  2. Jasmina Ferme
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Correspondence to Boštjan Brešar.

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Brešar, B., Ferme, J. Packing coloring of Sierpiński-type graphs. Aequat. Math. 92, 1091–1118 (2018). https://doi.org/10.1007/s00010-018-0561-8

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  • DOI: https://doi.org/10.1007/s00010-018-0561-8

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