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Edge-Colorings of 4-Regular Graphs with the Minimum Number of Palettes

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Abstract

A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Horňák, Kalinowski, Meszka and Woźniak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs. In particular, we show that certain values of the palette index imply the existence of an even cycle decomposition of size 3 (a partition of the edge-set of a graph into 3 2-regular subgraphs whose connected components are cycles of even length). This result can be extended to 4d-regular graphs. Moreover, in studying the palette index of a 4-regular graph, the following problem arises: does there exist a 4-regular graph whose even cycle decompositions cannot have size smaller than 4?

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Authors and Affiliations

  1. Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, via Amendola 2, 42122, Reggio Emilia, Italy

    Simona Bonvicini

  2. Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, via Campi 213/b, 41126, Modena, Italy

    Giuseppe Mazzuoccolo

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  1. Simona Bonvicini
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  2. Giuseppe Mazzuoccolo
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Correspondence to Simona Bonvicini.

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Bonvicini, S., Mazzuoccolo, G. Edge-Colorings of 4-Regular Graphs with the Minimum Number of Palettes. Graphs and Combinatorics 32, 1293–1311 (2016). https://doi.org/10.1007/s00373-015-1658-7

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  • DOI: https://doi.org/10.1007/s00373-015-1658-7

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