Face Labeling of Graph Embedded on the Surface of Klein Bottle

Abstract

We deal with the problem of labeling the Klein Grid’s Km n vertices, edges and faces with mn squares by the consecutive positive integers from 1 up to jV(Km n )j + jE(Km n )j + jF(Km n )j in such a way that if we add up all the labels of a 4-sided face that becomes the weight of that face. With common difference d these weight of the faces form an arithmetic progression. We will find out the existence of such type of labelings for several differences of d.