On a Relationship between Completely Separating Systems and Antimagic Labeling of Regular Graphs

Abstract

A completely separating system (CSS) on a finite set [n] is a collection C of subsets of [n] in which for each pair a ≠ b ∈ [n], there exist A, B ∈ C such that a ∈; A, b ∉ A and b ∈ B, a ∉ B. An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, ..., q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In this paper we show that there is a relationship between CSSs on a finite set and antimagic labeling of graphs. Using this relationship we prove the antimagicness of various families of regular graphs.