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

Nicolas Bousquet (1) , Antoine Dailly (2) , Eric Duchene (2) , Hamamache Kheddouci (2) , Aline Parreau (2)

1 G-SCOP_OC - Optimisation Combinatoire
2 GOAL - Graphes, AlgOrithmes et AppLications

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.

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Nicolas Bousquet, Antoine Dailly, Eric Duchene, Hamamache Kheddouci, Aline Parreau. A Vizing-like theorem for union vertex-distinguishing edge coloring. Discrete Applied Mathematics, 2017, 232, pp.88-98. ⟨10.1016/j.dam.2017.07.002⟩. ⟨hal-01313088v2⟩