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On the packing coloring of base-3 Sierpiński graphs and H-graphs

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Abstract

For a nondecreasing sequence of integers \(S=(s_1, s_2, \ldots )\) an S-packing k-coloring of a graph G is a mapping from V(G) to \(\{1, 2,\ldots ,k\}\) such that vertices with color i have pairwise distance greater than \(s_i\). By setting \(s_i = d + \lfloor \frac{i-1}{n} \rfloor \) we obtain a (dn)-packing coloring of a graph G. The smallest integer k for which there exists a (dn)-packing coloring of G is called the (dn)-packing chromatic number of G. In the special case when d and n are both equal to one we obtain the packing chromatic number of G. We determine the packing chromatic number of base-3 Sierpiński graphs and provide new results on (dn)-packing chromatic colorings for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for H-graphs.

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Acknowledgements

The authors acknowledge the financial support from the National Key R & D Program of China under grants 2017YFB0802300 and 2017YFB0802303, the National Natural Science Foundation of China under the grant 11361008, the Applied Basic Research (Key Project) of Sichuan Province under grant 2017JY0095 and the Slovenian Research Agency (research core funding No. P1-0297, projects J1-1693 and J1-9109).

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Correspondence to Aleksander Vesel.

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Deng, F., Shao, Z. & Vesel, A. On the packing coloring of base-3 Sierpiński graphs and H-graphs. Aequat. Math. 95, 329–341 (2021). https://doi.org/10.1007/s00010-020-00747-w

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  • DOI: https://doi.org/10.1007/s00010-020-00747-w

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