On $(k,d)$-Hooked Skolem Graceful Graphs

Abstract

A graph $(p,q)$ graph $G=(V,E)$ is said to be $(k,d)$-hooked Skolem graceful if there exists a bijection $f:V(G)\to \{1,2,\dots ,p-1,p+1\}$ such that the induced edge labeling $g_f:E\to \{k,k+d,\dots ,k+(n-1)d\}$ defined by $g_f(uv)=|f(u)-f(v)|$ $\forall uv\in E$ is also bijective, where $k$ and $d$ are positive integers. Such a labeling $f$ is called $(k,d)$-hooked Skolem graceful labeling of $G.$ Note that when $k=d=1$, this notion coincides with that of Hooked Skolem (HS) graceful labeling of the graph G. In this paper, we present some preliminary results on $(k,d)$-hooked Skolem graceful graphs and prove that $n{K}_2$ is $(2,1)$-hooked Skolem graceful if and only if $n \equiv 1~\mbox{or}~2(\bmod~ 4)$.

Journal Paper consists of 9 pages