On Antimagic Labeling of Odd Regular Graphs Tao-Ming Wang and Guang-Hui Zhang Department of Applied Mathematics Tunghai University Taichung, Taiwan 40704, R.O.C wang@go.thu.edu.tw Abstract. 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 K2 are antimagic. Another weaker version of the conjec- ture is every regular graph is antimagic except K2. 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 Zn are antimagic. Keywords: antimagic labeling, regular graph, perfect matching, 2-factor, generalized Petersen graph, Cayley graph, circulant graph. 1 Introduction All graphs in this paper are finite simple, undirected, and without loops unless otherwise stated. In 1990, N. Hartsfield and G. Ringel [9] introduced the concepts called antimagic labeling and antimagic graphs. Definition 1. For a graph G =(V,E) with q edges and without any isolated vertex, an antimagic edge labeling is a bijection f : E →{1, 2, ··· ,q}, such that the induced vertex sum f + : V → Z + given by f + (u)= ∑ {f (uv): uv ∈ E} is injective. A graph is called antimagic if it admits an antimagic labeling. N. Hartsfield and G. Ringel showed that paths, cycles, complete graphs K n (n ≥ 3) are antimagic. They conjectured that all connected graphs besides K 2 are antimagic, which remains unsettled. In 2004 N. Alon et al [1] showed that the last conjecture is true for dense graphs. They showed that all graphs with n(≥ 4) vertices and minimum degree Ω(log n) are antimagic. They also proved that if G is a graph with n(≥ 4) vertices and the maximum degree Δ(G) ≥ n − 2, then G is antimagic and all complete partite graphs except K 2 are antimagic. In 2005, T.-M. Wang [15] studied antimagic labeling of sparse graphs, and showed that the toroidal grid graphs are antimagic. In 2008, T.-M. Wang et al. [16] S. Arumugam and B. Smyth (Eds.): IWOCA 2012, LNCS 7643, pp. 162–168, 2012. c  Springer-Verlag Berlin Heidelberg 2012