Union vertex-distinguishing edge colorings

Abstract

The union vertex-distinguishing chromatic indexχ∪′(G) of a graph G is the smallest natural number k such that the edges of G can be assigned nonempty subsets of [k] so that the union of the subsets assigned to the edges incident to each vertex is different. We prove that χ∪′(G)∈{⌈log2⁡(n+1)⌉,⌈log2⁡(n+1)⌉+1} for any graph G on n vertices without a component of order at most 2. This answers a question posed by Bousquet, Dailly, Duchêne, Kheddouci and Parreau, and independently by Chartrand, Hallas and Zhang.