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.