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Coupon-Coloring and Total Domination in Hamiltonian Planar Triangulations

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Abstract

We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering the conjecture of Goddard and Henning.

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Acknowledgements

This work started at the Workshop on Graph and Hypergraph Domination in Balatonalmádi in June 2017. The author is thankful to the organizers and Michael Henning for posing the problem, and to Claire Pennarun, Dömötör Pálvölgyi, Balázs Keszegh, Zoltán Blázsik and Dániel Lenger for the discussion on the topic. Also, grateful acknowledgement is due to Claire Pennarun and the two anonymous referees for their helpful suggestions in order to improve the presentation of the paper.

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Authors and Affiliations

  1. MTA–ELTE Geometric and Algebraic Combinatorics Research Group, Budapest, Hungary

    Zoltán Lóránt Nagy

  2. Department of Computer Science, ELTE Eötvös Loránd University, Pázmány P. sétány 1/C, Budapest, 1117, Hungary

    Zoltán Lóránt Nagy

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  1. Zoltán Lóránt Nagy
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Correspondence to Zoltán Lóránt Nagy.

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The author is supported by the Hungarian Research Grant (OTKA) No. K 120154 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

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Nagy, Z.L. Coupon-Coloring and Total Domination in Hamiltonian Planar Triangulations. Graphs and Combinatorics 34, 1385–1394 (2018). https://doi.org/10.1007/s00373-018-1945-1

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  • DOI: https://doi.org/10.1007/s00373-018-1945-1

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