Labelings of type (1,1,1) for toroidal fullerenes

In this paper we deal with the problem of labeling the vertices, edges, and faces of a toroidal fullerene H n with mn hexagons by the consecutive integers from 1 up to j V (H n)j + j E(H n)j + j F(H n)j in such a way that the set of face-weights of 6-sided faces forms an arithmetic progression with common dierence d, where by face-weight we mean the sum of the label of that face and the labels of vertices and edges surrounding that face. The paper examines the existence of such labelings for several dierences d.