A Vizing-like theorem for union vertex-distinguishing edge coloring
  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. 
Nicolas Bousquet (G-SCOP), Antoine Dailly (GOAL), Eric Duchene (GOAL),   Hamamache Kheddouci (GOAL), Aline Parreau (GOAL)
cs.DM math.CO