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

Abstract

For any two positive integers \(m,n\), we denote the graph \(K_{m,n}\odot K_1\) by \(G\). Ma Ke-Jie proposed a conjecture [9] that pendant edge extension of a complete bipartite graph is a \(k\)-graceful graph for \(k \ge 2\). In this paper we prove his conjecture for \(n\le m < n^2+\lfloor\frac{k}{n}\rfloor+ r\).