Graceful labeling for open superstar of complete bipartite graphs S2(t·Km,n)

Abstract

A graceful labeling of graph G with m edges is a one-to-one function f from vertices of G to the set {0, 1, 2, · · ·, m} such that the induced function f* from edges of G to the set {1, 2, 3, · · ·, m}, which is defined by the difference of its incident vertices labels, is a bijection. An open superstar of complete bipartite graphs is an integration of complete bipartite graphs and superstar graphs. A superstar graph is a graph obtained by replacing each edge of star St by a path P3 of three vertices. An open superstar of complete bipartite graph, S2(t·Km,n), is a graph obtained by identifying each pendant vertex of superstar with exactly one vertex of a complete bipartite graph Km,n. In this paper, we have given a graceful labeling for the open superstar of complete bipartite graphs S2(t·Km,n) for all positive integers m, n, and odd positive integers t.