On k-graceful labeling of pendant edge extension of complete bipartite graphs

Abstract

For any two positive integers m, n, we denote the graph Km,n ⊙ K1 by G. Ma Ke-Jie proposed a conjecture [9]

that pendant edge extension of a complete bipartite graph is a k- graceful graph for k > 2. In this paper we prove his conjecture for

n 6 m < n2 + ⌊ k n ⌋ + r.