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 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. © 2011 Springer-Verlag.