Vertex magic total labeling of products of cycles

A vertex magic total labeling of a graph G(V, E) assigns to all vertices and edges of G labels from the set {1, 2, ..., |V| + |E|} so that the sum (called the weight) of the vertex label and of labels of all adjacent edges does not depend on the vertex. A generalized (s, d)-vertex antimagic total labeling of G assigns positive integers to all vertices and edges of G so that the vertex weights form an arithmetic progression s,s + d, ..., s + (|V| - 1)d. We present a construction of vertex magic total labelings of products of cycles C-m x C-n for m,n >= 3, n odd. The construction is based on a generalized (s, d)-vertex antimagic labeling of cycles in which non- consecutive integers are used.