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.

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 $n \equiv 1~\mbox{or}~2(\bmod~ 4)$.

Equations

$$\sum _{i=1}^n(b_i-a_i)=k+(k+d)+\cdots +k+(n-1)d$$

$$\begin{array}{rl}\sum _{i=1}^n{b}_i+\sum _{i=1}^n{a}_i& =1+2+\cdots +(2n-1)+(2n+1)\\ & =n(2n-1)+(2n+1)\end{array}$$

$$\sum _{i=1}^n{b}_i=nk+2n^2+n+1+\frac{n(n-1)}{2}d$$

$$f(a_i)=\{\begin{array}{ll}i,& \text{for i=1,2;}\\ i+1,& \text{for 3≤i≤2r−2;}\\ \frac{n+4}{2},& \text{for i=2r−1;}\\ \frac{3n+2}{4},& \text{for i=2r;}\\ \frac{n-4}{2}+i,& \text{for 2r+1≤i≤n.}\end{array}$$

$$f(b_i)=\{\begin{array}{ll}3,& \text{for i=1;}\\ \frac{n+2}{2},& \text{for i=2;}\\ n+2-i,& \text{for 3≤i≤r;}\\ n+1-i,& \text{for r+1≤i≤2r−3;}\\ \frac{3n}{2},& \text{for i=2r−2;}\\ \frac{3n-2}{2},& \text{for i=2r−1;}\\ \frac{7n-2}{4},& \text{for i=2r;}\\ 2n+1,& \text{for i=2r+1;}\\ \frac{5n+4}{2}-i,& \text{for 2r+2≤i≤3r;}\\ \frac{5n+2}{2}-i,& \text{for 3r+1≤i≤n.}\end{array}$$

$$f(a_i)$$

$$f(b_i)$$

Full text

2010 MATHEMATICS SUBJECT CLASSIFICATION:

**05C 78.

**

KEY WORDS: Hooked sequence, hooked Skolem graceful graphs

1 Introduction

By a graph $G=(V,E)$, we mean a finite undirected graph without loops or multiple edges. The order $|V|$ and the size $|E|$ of $G$ are denoted by $p$ and $q$ respectively. For graph theoretic terminology and notations we refer to West [10].

While studying the structure of Steiner triple systems, Skolem [9] consid- ered the following problem: Is it possible to distribute the numbers $1,2,\dots,2p$ into $n$ pairs $(a_{i},b_{i})$ such that we have $b_{i}-a_{i}=i$ for $i=1,2,\dots,p$? In the sequel, a set of pairs of this kind is called $1,+1$ system because the difference $b_{i}-a_{i}$ begins with $1$ and increases by $1$ when $i$ increases by $1$. Skolem [9] proved that a $1,+1$ system exists if and only if $p\equiv 0~{}\mbox{or}~{}1(\bmod~{}4)$. A $1,+1$ system is also known as Skolem sequence, which is defined as follows.

Definition 1.1**.**

Let $\langle C_{i}\rangle$ be a sequence of $2p$ terms, where $1\leq C_{i}\leq p$. If each number $i$ occurs exactly twice in the sequence and $|j_{2}-j_{1}|=i$ if $i=C_{j_{1}}=C_{j_{2}}$ then $\langle C_{i}\rangle$ is called a Skolem sequence.

This concept was used by Lee and Shee [4] to introduce the notion of Skolem gracefulness of graphs.

Definition 1.2**.**

A Skolem graceful labeling of a $(p,q)$ graph $G=(V,E)$ is a bijection $f:V\rightarrow\{1,2,\dots,p\}$ such that the induced labeling $g_{f}:E\rightarrow\{1,2,\dots,q\}$ defined by $g_{f}(uv)=|f(u)-f(v)|$ is also a bijection. If such a labeling exists then the graph $G$ is called a Skolem graceful graph.

If a graph $G$ with $p$ vertices and $q$ edges, is graceful then $q\geq p-1$, while if it is Skolem graceful, then $q\leq p-1$. Thus, as noted in [4], Skolem graceful labelings nearly complement graceful labelings, and a graph with $q=p-1$ is graceful if and only if it is Skolem graceful.

O’Keefe [2] extended the methods of Skolem sequences for $k\equiv 2~{}\mbox{or}~{}3(\bmod~{}4)$ by showing that the numbers $1,2,\dots,2k-1,2k+1$ can be distributed into $k$ disjoint pairs $(a_{i},b_{i})$ such that $b_{i}=a_{i}+i$ for $i=1,2,\dots,k$.

Motivated by this, Shalaby [5] defined the notion of hooked Skolem sequences as follows.

Definition 1.3**.**

A hooked Skolem sequence (HS) of order $k$ is a sequence $(c_{1},c_{2},\dots,c_{2k+1})$ of $2k+1$ integers satisfying the following conditions:

For every $r\in\{1,2,\dots,k\}$ there exist exactly two elements $c_{i}$ and $c_{j}$ such that $c_{i}=c_{j}=r$. 2. 2.

If $c_{i}=c_{j}=r$ with $i<j$, then $j-i=r$. 3. 3.

$c_{2n}=0$.

In [7] a hooked sequence is defined as a sequence $\{d,d+1,\dots,d+m-1\}$ for which there is a partition of the set $\{1,2,\dots,2m-1,2m+1\}$ into $m$ pairs $(a_{i},b_{i})$ such that the $m$ numbers $b_{i}-a_{i}$, $1\leq i\leq m$ are all of the integers $d,d+1,\dots,d+m-1$. Where $a_{i}$ and $b_{i}$ are interpreted as the two positions in the sequence where $b_{i}-a_{i}$ appears. For example $48574365387*6$ and $64758463573*8$ are hooked sequences where $d=3$ and $m=6$.

In [8] a hooked Skolem graceful graph is defined as follows: A $(p,q)$ graph $G=(V,E)$ is said to be 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 $g_{f}:E\rightarrow\{1,2,\dots,q\}$ defined by $g_{f}(uv)~{}=~{}|f(u)-f(v)|,~{}\forall uv\in E$ is also bijective. Such a labeling $f$ is called hooked Skolem graceful labeling of $G$.

In this paper, we introduce the notion of $(k,d)$-hooked Skolem graceful graph as follows:

Definition 1.4**.**

A $(p,q)$ 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 $g_{f}:E\rightarrow\{k,k+d,k+2d,\dots,k+(q-1)d\}$ 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 $G$.

2 Main Results

It follows from the definition that if $G$ is $(k,d)$-hooked Skolem graceful, then $q\leq p-1$. For any two disjoint subsets $A$ and $B$ of $V$, we denote by $m(A,B)$ the number of edges of $G$ with one end in $A$ and the other end in $B$.

Theorem 2.1**.**

Let $k$ and $d$ be two positive integers which are not simultaneously even. If $G$ is $(k,d)$-hooked Skolem graceful, then $V(G)$ can be partitioned into two subsets $V_{o}$ and $V_{e}$ satisfying the following conditions.

$m(V_{o},V_{e})=\lfloor\frac{q+1}{2}\rfloor$* if $k$ and $d$ are both odd.* 2. 2.

$m(V_{o},V_{e})=\lfloor\frac{q}{2}\rfloor$* if $k$ is even and $d$ is odd.* 3. 3.

$m(V_{o},V_{e})=q$* if $k$ is odd and $d$ is even.*

Proof.

Let $f$ be a $(k,d)$-hooked Skolem graceful labeling of $G$. Let $V_{o}=\{u\in V(G):f(u)\ \mbox{is odd}\}$ and $V_{e}=V(G)-V_{o}$. then $g_{f}(G)$ is odd if and only if the corresponding edge joins a vertex of $V_{o}$ and a vertex of $V_{e}$ and hence the result follows. ∎

In the following theorem we investigate the existence of $(k,d)$-hooked Skolem graceful labeling for $nK_{2}$.

Theorem 2.2**.**

If $nK_{2}$ is $(k,d)$-hooked Skolem graceful, then one of the following holds.

$n\equiv 1(\bmod~{}4)$, then $k$ is even. 2. 2.

$n\equiv 2(\bmod~{}4)$, then $d$ is odd. 3. 3.

$n\equiv 3(\bmod~{}4)$, then both $k$ and $d$ are even or they are odd.

Proof.

Let $f$ be a $(k,d)$-hooked Skolem graceful labeling of $nK_{2}$. Let $e_{i}=u_{i}v_{i}$ be the components of $nK_{2}$ and let $f(u_{i})=a_{i}$, $f(v_{i})=b_{i}$ and $b_{i}>a_{i}$, $1\leq i\leq n$. Since the set of vertex labels is $\{1,2,\dots,2n-1,2n+1\}$ and the set of edge labels is $\{k,k+d,\dots,k+(n-1)d\}$, we have

[TABLE]

[TABLE]

On adding (1) and (2) we have,

[TABLE]

If $n\equiv 1(\bmod~{}4)$, then $2n^{2}+n+1+\frac{n(n-1)}{2}$ is even and hence $k$ is even. Hence condition 1 holds. By a similar argument conditions 2 and 3 can be proved. ∎

Corollary 2.3**.**

The necessary conditions for the sequence $\{d,d+1,d+2,\dots,d+m-1\}$ to be hooked are:

$m(m+1-2d)+2\geq 0$* and* 2. 2.

$m\equiv 2~{}\mbox{or}~{}3(\bmod~{}4)$* for $d$ odd and $m\equiv 2~{}\mbox{or}~{}1(\bmod~{}4)$ for $d$ even.*

The following theorem gives the necessary and sufficient condition for $nK_{2}$ to be $(2,1)$-hooked Skolem graceful.

Theorem 2.4**.**

The graph $nK_{2}$ is $(2,1)$-hooked Skolem graceful if and only if $n\equiv 1~{}\mbox{or}~{}2(\bmod~{}4)$.

Proof.

If $nK_{2}$ is $(2,1)$-hooked Skolem graceful, then it follows from Theorem 2.2 that $n\equiv 1~{}\mbox{or}~{}2(\bmod~{}4)$.

Conversely, let $n\equiv 1~{}\mbox{or}~{}2(\bmod~{}4)$. Let $e_{i}=a_{i}b_{i}$ be the edges of $nK_{2}$ with $b_{i}>a_{i}$, $1\leq i\leq n.$

Case 1: $n\equiv 2(\bmod~{}4)$.

Let $n=4r-2$, where $r$ is a positive integer. For $r=1,~{}2$ and $3$, the $(2,1)$-hooked Skolem graceful labeling of $2K_{2}$, $6K_{2}$ and $10K_{2}$ are given in Figure 1.

For $r\geq 4$, we define the vertex labeling $f$ as follows:

[TABLE]

[TABLE]

Case 2: $n\equiv 1(\bmod~{}4).$

Let $n=4r-3$, where $r$ is a positive integer. For $r=1$ and $2$ the $(2,1)$-hooked Skolem graceful labelings of $K_{2}$ and $5K_{2}$ are given in Figure 2.

For $r\geq 3$, we define the vertex labeling $f$ as follows:

[TABLE]

[TABLE]

In each case, it can be easily verified that the induced edge function $g_{f}$ defined by $g_{f}(e_{i})=|b_{i}-a_{i}|$ has the required properties to qualify $f$ to be a $(2,1)$-hooked Skolem graceful labeling of $nK_{2}$ and the cases exhaust all the possibilities. This completes the proof. ∎

3 Conclusion and Scope

In this paper, we have introduced the notion of $(k,d)$-hooked Skolem graceful graphs and observe that $k=d=1$ coincides with the notion of hooked Skolem graceful labeling of a graph $G$. We have given some necessary or sufficient conditions for a graph $G$ to be $(k,d)$-hooked Skolem graceful. We have proved that $nK_{2}$ is $(2,1)$-hooked Skolem graceful if and only if $n\equiv 1~{}\mbox{or}~{}2(\bmod~{}4)$. Determining the value of $n$ for which $nK_{2}$ is $(k,d)$-hooked Skolem graceful for given values of $k$ and $d$ is an open problem.