Face antimagic labelings of toroidal and Klein bottle grid graphs

Abstract

In this paper we deal with the problem of labeling the vertices, edges and faces of a toroidal Tnm and Klein bottle Knm grid graphs with mn 4-sided faces by the consecutive integers from 1 up to |V(Tnm)| + |E(Tnm)| + |F(Tnm)| and |V(Knm)| + |E(Knm)| + |F(Knm)| in such a way that the label of a 4-sided face and the labels of the vertices and edges surrounding that face all together add up to a weight of that face. These face-weights then form an arithmetic progression with common difference d. The paper examines the existence of such labelings for several differences d.