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 (d, n)-packing coloring of a graph G. The smallest integer k for which there exists a (d, n)-packing coloring of G is called the (d, n)-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 (d, n)-packing chromatic colorings for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for H-graphs.
Similar content being viewed by others
References
Balogh, J., Kostochka, A., Liu, X.: Packing chromatic number of subcubic graphs. Discrete Math. 341, 474–483 (2018)
Brešar, B., Ferme, J.: Packing coloring of Sierpiński-type graphs. Aequationes Math. 92, 1091–1118 (2018)
Brešar, B., Ferme, J., Klavžar, S., Rall, D.F.: A survey on packing colorings (manuscript)
Brešar, B., Klavžar, S., Rall, D.F., Wash, K.: Packing chromatic number under local changes in a graph. Discrete Math. 340, 1110–1115 (2017)
Brešar, B., Klavžar, S., Rall, D.F.: On the packing chromatic number of Cartesian products, hexagonal lattice, and trees. Discrete Appl. Math. 155, 2303–2311 (2007)
Brešar, B., Klavžar, S., Rall, D.F.: Packing chromatic number of base-3 Sierpiński graphs. Graphs Comb. 32, 1313–1327 (2016)
Gastineau, N., Kheddouci, H., Togni, O.: Subdivision into \(i\)-packings and \(S\)-packing chromatic number of some lattices. Ars Math. Contemp. 9, 331–354 (2015)
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)
Goddard, W., Xu, H.: The \(S\)-packing chromatic number of a graph. Discuss. Math. Graph Theory 32, 795–806 (2012)
Jakovac, M., Klavžar, S.: Vertex-, edge-, and total-colorings of Sierpiński -like graphs. Discrete Math. 309, 1548–1556 (2009)
Klavžar, S., Vesel, A.: Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of (2,1)-colorings and independence numbers. Discrete Appl. Math. 129, 449–460 (2003)
Klavžar, S., Zemljič, S.S.: On distances in Sierpiński graphs: almost-extreme vertices and metric dimension. Appl. Anal. Discrete Math. 7, 72–82 (2013)
Korže, D., Vesel, A.: \((d, n)\)-packing colorings of infinite lattices. Discrete Appl. Math. 237, 97–108 (2018)
Korže, D.: A. Vesel, Packing coloring of generalized Sierpiński graphs, Discrete Math. Theor. Comput. Sci. 21 (2019) paper #7, 18 pp
Laïche, D., Sopena, É.: Packing colouring of some classes of cubic graphs. Australas. J. Combin. 72, 376–404 (2018)
Shao, Z., Vesel, A.: Modeling the packing coloring problem of graphs. Appl. Math. Model. 39, 3588–3595 (2015)
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-020-00747-w