On Antimagic Labeling of Odd Regular Graphs

An antimagic labeling of a finite simple undirected graph with q edges is a bijection from the set of edges to the set of integers {1, 2, ⋯ , q} such that the vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of labels of all edges incident to such vertex. A graph is called antimagic if it admits an antimagic labeling. It was conjectured by N. Hartsfield and G. Ringel in 1990 that all connected graphs besides K 2 are antimagic. Another weaker version of the conjecture is every regular graph is antimagic except K 2. Both conjectures remain unsettled so far. In this article, certain classes of regular graphs of odd degree with particular type of perfect matchings are shown to be antimagic. As a byproduct, all generalized Petersen graphs and some subclass of Cayley graphs of ℤ n are antimagic.