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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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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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