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arXiv · 1605.02588

A Vizing-like theorem for union vertex-distinguishing edge coloring

Nicolas Bousquet·Antoine Dailly·Eric Duchene·Hamamache Kheddouci·Aline Parreau
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Abstract

We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the problem of finding a coloring with the minimum number of colors where every vertex receives a distinct label. Finding such a coloring generalizes several other well-known problems of vertex-distinguishing colorings in graphs.We show that for any graph (without connected component reduced to an edge or a single vertex), the minimum number of colors for which such a coloring exists can only take 3possible values depending on the order of the graph. Moreover, we provide the exact value for paths, cycles and complete binary trees.

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Nicolas Bousquet, Antoine Dailly, Eric Duchene, Hamamache Kheddouci, Aline Parreau. 2016-05-09. A Vizing-like theorem for union vertex-distinguishing edge coloring. https://doi.org/10.1016/j.dam.2017.07.002

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