Products of Regular Graphs are Antimagic

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [5] conjectured that every simple connected graph, but K2, is antimagic. We prove that the Cartesian product graphs G1 ×G2 (where G1 is a connected k1-regular graph and G2 is a graph with maximum degree at most k2, minimum degree at least one) are antimagic, provided that k1 is odd and k 2 1 −k1 2 ≥ k2, or, k1 is even and k 2 1 2 ≥ k2 and k1, k2 are not both equal to 2. By combining the above result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [9], we obtain that all Cartesian products of two or more regular graphs are antimagic. We also give a generalization of the above antimagicness result on G1×G2, for which G1 is not necessarily connected.