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

A graph $(p,q)$ graph $G=(V,E)$ is said to be $(k,d)$-hooked Skolem graceful if there exists a bijection $f:V (G)\rightarrow {1, 2, \dots, p-1, p+1}$suchthattheinducededgelabeling$g_f : E \rightarrow {k, k+d, \dots, k+(n-1)d }$definedby$g_f (uv) = |f(u) - f(v)|$ $\mathrm{\forall }uv\in E$ is also bijective, where $k$ and $d$ are positive integers. Such a labeling $f$ is called $(k,d)$-hooked Skolem graceful labeling of $G\mathrm{.}$ 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 $(k, d)$-hookedSkolemgracefulgraphsandprovethat$nK_2$is$(2, 1)$-hooked Skolem graceful if and only if $n\equiv 1or2(4)$.