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 T n m and Klein bottle K n m grid graphs with m n 4-sided faces by the consecutive integers from 1 up to | V ( T n m ) | + | E ( T n m ) | + | F ( T n m ) | and | V ( K n m ) | + | E ( K n m ) | + | F ( K n m ) | 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 .