Lattice grids and prisms are antimagic

An antimagic labelling 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 labelling. In 1990, Hartsfield and Ringel conjectured that every connected graph, but K2, is antimagic. In [T.-M. Wang, Toroidal grids are antimagic, in: Proc. 11th Annual International Computing and Combinatorics Conference, COCOON’2005, in: LNCS, vol. 3595, Springer, 2005, pp. 671–679], Wang showed that the toroidal grids (the Cartesian products of two or more cycles) are antimagic. Two open problems left in Wang’s paper are about the antimagicness of lattice grid graphs and prism graphs, which are the Cartesian products of two paths, and of a cycle and a path, respectively. In this article, we prove that these two classes of graphs are antimagic, by constructing such antimagic labellings.