Discovery of Some New Classes of Graceful Unicyclic Graphs

Abstract

Graceful graphs were first studied by Rosa [17]. A graceful labeling $f$ of a graph G is a one-to-one map from the set of vertices of $G$ to the set {0.1,., |E(G)|}. where for edges $xy$, the induced edge labels |f(x) – f (y)| form the set {1,2,., |E(G)|, with no label repeated. In this paper, we investigate the set of labels taken by the central vertex of the star in the graph $K_{1.m-1}\oplus C_n$, for each graceful labeling. We also study gracefulness of certain unicyclic graphs where paths $P_3,P_2$ are pendant at vertices of the cycle. For these unicyclic graphs, the deletion of any edge of the cycle does not result in a caterpillar.