Toroidal Grids Are Anti-magic

Abstract

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the integers {1,..., q} such that all p vertex sums are pairwise distinct, where the vertex sum on a vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it has an anti-magic labeling. Hartsfield and Ringel [3] conjectured that all connected graphs except K2 are anti-magic. Recently, N. Alon et al [1] showed that this conjecture is true for p-vertex graphs with minimum degree Ω(log p). They also proved that complete partite graphs except K2 and graphs with maximum degree at least p–2 are anti-magic. In this article, some new classes of anti-magic graphs are constructed through Cartesian products. Among others, the toroidal grids Cm × Cn(the Cartesian product of two cycles), and the higher dimensional toroidal grids $C_{m_1} \times C_{m_2} \times ...... \times C_{m_t}$, are shown to be anti-magic. Moreover, the more general result is also proved to be true: H × Cn (hence Cn × H) is anti-magic, where H is an anti-magic k-regular graph, where k>1.