Graceful labeling on a multiple-fan graph with pendants

Abstract

An injective function f from the set of vertices in a graph G to a set {0,1,…,n} is called graceful labeling if the function f induced the edge function f* from the set of edges of G to a set of positive integers {1,2,…,n} with f*(xy) = |f(x) − f(y)| for every edge xy ∈ E(G) such that every edge has a different label. A graph G that can be labeled with graceful labeling is called a graceful graph. The multiple-fan graph with k pendants F(n1,n2,…,nr),k is a graph obtained from r fan graphs Fni, with i = 1,2,…,r, which share the center vertex and then we add k pendants to the center vertex of the multiple-fan graph. In this paper, we show that the multiple-fan graph with k pendants is graceful, with ni ≥ 2, i = 1,2,…, r and k ≥ r − 1.