ON k-GRACEFUL DIGRAPHS

In this paper we extend the idea of k- graceful labeling of undirected graphs to a directed graphs: A simple directed graph D with n vertices and e edges is labeled by assigning each vertex a distinct element from the set Z(e+k) = {0, 1, 2, ... , e + k - 1}, where k is a positive integer and an edge xy from vertex x to vertex y is labeled with theta(x, y) = theta(y) - theta(x)mod(e + k), where theta(y) and theta(x) are the values assigned to the vertices y and x respectively. A labeling is a k - graceful labeling if all theta(x, y) are distinct and belong to {k, k + 1, ... , k + e - 1}. If a digraph D admits a k- graceful labeling then D is a k - graceful digraph. We also provide a list of values of k for which the unidirectional cycle (C) over right arrow (n) admits a k- graceful labeling. Further, we give a necessary and sufficient condition for the outspoken unicyclic wheel to be k- graceful and prove that to provide a list of values of k > 1, for which the unicyclic wheel (W) over right arrow (n) is k- graceful is NP - complete.