On gracefulness in the path union and cycle of $P_m \theta S_n$

Abstract

Abstract:  A   graceful labeling  of a graph $G$ with $q$ edges is an injection \(f:V(G) \to \{ 0,1,2,...,q\} \) with the property that the resulting edge labels are also distinct, where an edge incident with the vertices $u$ and $v$ is assigned the label \(\left| {f\left( u \right)-f(v)} \right|\). A graph which admits a graceful labeling is called a graceful graph.  In this paper, we prove that the path union of isomorphic copies of $P_m \theta S_n$ by fixing the middle vertex of the graph as the root  is graceful and the graphs obtained by joining each vertex of cycle $C_k$ with the middle vertex of the isomorphic copies of the $P_m \theta S_n$ are graceful where $m \equiv 1 \left(\bmod 2 \right)$ and $k \equiv 0,3 \left(\bmod 4 \right)$.