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

A vertex-distinguishing coloring of a graph G consists in an edge or a vertex coloring (not necessarily proper) of G leading to a labeling of the vertices of G, where all the vertices are distinguished by their labels. There are several possible rules for both the coloring and the labeling. For instance, in a set irregular edge coloring [5], the label of a vertex is the union of the colors of its adjacent edges. Other rules for the labeling of a vertex from an edge coloring have been studied: the multiset of its adjacent colors [1], their sum [3], product or difference [6] (for those three rules, the colors must be integers)... The variant where the edge coloring is proper has also been studied [2]. If the vertices are colored, we can define the identifying coloring [4], in which each vertex is assigned a label corresponding to the union of its closed neighbourhood colors. Motivated by a generalization of the set irregular edge coloring problem, we introduce a variant of the problem: to each edge is associated a nonempty set of colors. Given a simple graph G, a k-coloring of G is a function f : E(G)→ 2{1...,k} where every edge is labeled using a non-empty subset of {1, . . . , k}. For any k-coloring f of G, we define, for every vertex u, the set idf (u) as follows: idf (u) = ⋃