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

## TL;DR

This paper introduces the concept of $(k,d)$-hooked Skolem graceful graphs, generalizing previous notions, and characterizes when disjoint unions of edges are $(2,1)$-hooked Skolem graceful based on modular conditions.

## Contribution

It defines the $(k,d)$-hooked Skolem graceful labeling and provides a characterization for when multiple edges form such graphs.

## Key findings

- $nK_2$ is $(2,1)$-hooked Skolem graceful iff $n
ot\equiv 0, 3 mod 4$
- Introduces a generalized graph labeling concept
- Provides initial results on properties of these graphs

## Abstract

A graph $(p, q)$ graph $G = (V, E)$ is said to be $(k, d)$-hooked Skolem graceful if there exists a bijection $f:V (G)\rightarrow \{1, 2, \dots, p-1, p+1\}$ such that the induced edge labeling $g_f : E \rightarrow \{k, k+d, \dots, k+(n-1)d \}$ 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 $nK_2$ is $(2, 1)$-hooked Skolem graceful if and only if $n \equiv 1~\mbox{or}~2(\bmod~ 4)$.