Research Article | Open Access

Mohamed R. Zeen El Deen, "Edge *δ*− Graceful Labeling for Some Cyclic-Related Graphs", *Advances in Mathematical Physics*, vol. 2020, Article ID 6273245, 18 pages, 2020. https://doi.org/10.1155/2020/6273245

# Edge *δ*− Graceful Labeling for Some Cyclic-Related Graphs

**Academic Editor:**Ivan Giorgio

#### Abstract

In this paper, we introduce a new type of labeling of a graph with vertices and edges called edge *δ*− graceful labeling, for any positive integer , as a bijective mapping of the edge set into the set such that the induced mapping , given by , where , is an injective function. We prove the existence of an edge *δ*− graceful labeling, for any positive integer , for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph , the prism graph , the prism of the wheel , the gear graph , the closed helm CH* _{n}*, the butterfly graph , and the friendship Fr

*.*

_{n}#### 1. Introduction

The graphs considered here will be finite, undirected, and simple where and will denote the vertex set and edge set of a graph , respectively, and .

A labeling of a graph is a mapping that carries graph elements (edges or vertices, or both) to positive integers, subject to certain constraints. A labeling of a graph is called edge labeling if the domain of the mapping is the edge set. Graph labeling methods are used for application problems in a communication network addressing system, for fasting communication in sensor networks, and for designing fault-tolerant systems with facility graphs, in coding theory for the design of good radar type codes, and can also be used for issues in mobile ad hoc networks [1–3].

In the early 1960s, the idea of graph labelings was introduced by Rosa in [4]. Following this paper, different techniques have been studied in graph labelings. One such graph labeling technique is the edge graceful labeling introduced by Lo [5]. In 1985 as a bijective, from the set of edges to the set such that the induced map from to given by is a bijective. The graph that admits a graceful labeling is called a graceful graph.

Solairaju and Chithra [6] in 2009 introduced a labeling of called edge odd graceful labeling, which is a bijection from the set of edges to the set such that the induced map from to given by is an injective. For many results on this type of labeling, see [7–10].

Recently, a new type of labeling is introduced by Elsonbaty and Daoud [11] called edge even graceful labeling, which is a bijective from the set of edges to the set such that the induced map from to given by where is an injective. Several results have been published on edge even graceful labeling, see [12–15]. For a detailed survey on graph labeling, refer to a dynamic survey of graph labeling [16].

Now, we introduce a generalization of the edge graceful labeling to edge *δ*− graceful labeling for any positive integer .

*Definition 1. *An edge *δ*− graceful labeling of a graph , with vertices and edges, for any positive integer , is a bijective mapping of the edge set into the set such that the induced mapping , given by , where , is an injective function. The graph that admits an edge *δ*− graceful labeling is called an edge *δ*− graceful graph.

Note that, if , we have the edge even labeling; also, if , we have the edge triple labeling and so on. The odd cycle whose vertices and edges () where is an edge *δ*− graceful graph if and only if is odd. Labeling the edges, respectively, by yields the labels on vertices, respectively. In Figure 1, we present an edge 5− graceful labeling of a graph and an edge 4− graceful labeling of another graph.

#### 2. Edge *δ*− Graceful Labeling of the Wheel Graph

Theorem 2. *For any positive integer , the wheel graph , is an edge δ− graceful graph.*

*Proof. *Let be the vertices of with hub vertex , and the edges of will be . So, and . There are three cases:

*Case 1. *, or .

We define the labeling function as follows:
Then, the induced vertex labels are
Hence, the labels of the vertices will be , respectively, which are all a multiple of and distinct numbers.
(i)If , then
(ii)If , then

*Case 2. *. Define the labeling function as follows:
Then, the induced vertex labels are and . Hence, the labels of the vertices will be , respectively.
(i)If , then

*Case 3. *. Define the labeling function as follows:
Then, the induced vertex labels are , , and , . Hence, the labels of the vertices will be , respectively.
(i)If , then
It is clear that, in all cases, for all , the labels of the vertices are all distinct, multiple of , and different from which complete the proof.

It should be noted that is not an edge *δ*− graceful graph because for any bijective function , there is no injective induced function that satisfied the requirements

*Illustration 1*. In Figure 2, we present with edge 5− graceful labeling, with edge 3− graceful labeling, with edge 4− graceful labeling, and with edge 6− graceful labeling, respectively.

#### 3. Edge *δ*− Graceful Labeling of the Alternate Triangular Cycle

*Definition 3 (see [16]). *An alternate triangular cycle is obtained from an even cycle by joining and to a new vertex . That is, every alternate edge of a cycle is replaced by .

Theorem 4. *For any positive integer , the alternate triangular cycle is an edge δ− graceful graph.*

*Proof. *Let the alternate triangular cycle be given as in Figure 3 with the edge set will be . The graph has and . We define the labeling function by
By using the above labeling pattern, the induced vertex labels will be
Hence, the labels of the vertices will be , respectively, the labels of the vertices will be , respectively, and the labels of the vertices will be , respectively. Hence, there are no repeated vertex labels which complete the proof.

*Illustration 2*. In Figure 4, we present an alternate triangular cycle with an edge 4− graceful labeling.

#### 4. Edge *δ*− Graceful Labeling of the Double Wheel Graph

Theorem 5. *For any positive integer , the double wheel graph is an edge δ− graceful graph.*

*Proof. *The double wheel graph consists of two cycles of vertices connected to common hub. Let be the vertices of one wheel and be the vertices of other wheel with hub vertex ; the edges of will be . So, and , see Figure 5. There are two cases:

*Case 1. *When is even. We define the labeling function as follows:
Then, the induced vertex labels are
Hence, the labels of the vertices are , respectively, and the labels of the vertices are , respectively.
Since , then

*Case 2. *If is odd number, define the labeling function as follows:
Then, the induced vertex labels are
Hence, the labels of the vertices will be , respectively, and the labels of the vertices are , respectively.
Clearly, the vertex labels are all distinct, multiple of *δ*, and different from . Thus, the double wheel graph is an edge *δ*− graceful graph for any positive integer *δ*.

*Illustration 3*. In Figure 6, we present with edge 4− graceful labeling and with edge 5− graceful labeling.

#### 5. Edge *δ*− Graceful Labeling of the Prism Graph

Theorem 6. *For any positive integer δ, the prism graph is an edge δ− graceful graph.*

*Proof. *Let be the vertices of and be a copy of . The prism is defined by joining each of to the corresponding vertex of for all ; edges of will be . Thus, an n-prism graph has vertices and edges.

Let the vertex and edge symbols be given as in Figure 7.

Define the labeling mapping by
So the induced vertex labels will be
Hence, the labels of the vertices will be , respectively, and the labels of the vertices will be , respectively.

Overall, the labels of the vertices are all different and multiplies of . Thus, the prism graph is an edge *δ*− graceful graph for any positive integer .

*Illustration 4*. In Figure 8, we present with edge 7− graceful labeling.

#### 6. Edge *δ*− Graceful Labeling of the Prism of the Wheel

*Definition 7 (see [3]). *For , let be the vertices of with hub vertex and be a copy of . Define , called the *prism* of , by , i.e., joining of to the corresponding vertex of and each of to the corresponding vertex of for all . Thus, .

Theorem 8. *For any positive integer , the prism of the wheel is an edge δ− graceful graph, when is even.*

*Proof. *The prism of the wheel has vertices and edges, the edges will be . Let the vertex and edge symbols be given as in Figure 9.

We define the labeling function by
In view of the above labeling pattern, then the induced vertex labels are
Hence, the labels of the vertices are , respectively, and the labels of the vertices are , respectively.

Hence, there are no repeated vertex labels. Also in this labeling, we can check that
(i)(ii)Then, , finally
There are three cases:

*Case 1. *When or or .
(i)If , then (ii)If , then (iii)If , then

*Case 2. *When , then