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

TL;DR

This paper studies a new vertex-distinguishing edge coloring problem where edges are assigned color sets, and each vertex's label is the union of incident edge colors, generalizing several known problems.

Contribution

It establishes that the minimum number of colors needed is limited to three possible values depending on the graph's order and provides exact values for specific graph classes.

Findings

Minimum colors depend on graph order, with only three possible values.

Exact color counts are determined for paths, cycles, and binary trees.

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.