M Modulo N graceful labeling of path union and join sum of complete bipartite graphs with its algorithms

A graceful labeling of a graph G(p, q) is an injective assignment of labels from the set {0, 1, ..., q} to the vertices of G such that when each edge of G has been assigned a label defined by the absolute difference of its end-vertices, the resulting edge labels are distinct. In this paper we used the new labeling technique known as M modulo N graceful labeling and prove that path union of complete bipartite graphs and join sum of complete bipartite graphs are M modulo N graceful labeling. We also give a C + + program for finding M modulo N graceful labeling on above said graphs

γ-labelings of complete bipartite graphs

Explicit formulae for the γ-min and γ-max labeling values of complete bipartite graphs are given, along with γ-labelings which achieve these extremes. A recursive formula for the γ-min labeling value of any complete multipartite is also presented