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 CHn, the butterfly graph , and the friendship Frn.
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
In this case, we have will equal to when , so we change the labeling of two edges () and () as follows: and .
Then, , and .
Case 3. When , then
In this case, we have will equal to when , so we change the labeling of two edges () and as follows: and .
Then, , and .
In all cases, and are not congruent to nor which completes the proof.
Illustration 5. In Figure 10, we present with an edge 3− graceful labeling and with an edge 4− graceful labeling.
Theorem 9. For any positive integer δ, the prism of the wheel is an edge δ− graceful graph, when is odd.
Proof. Let the vertex and edge symbols be given as in Figure 9. Define the labeling function by
In view of the above labeling pattern, we can check that
(i)(ii)(iii) then, the induced vertex labels are
Hence, the labels of the vertices are and the labels of the vertices are , respectively. Also, the labels of the vertices are , and the labels of the vertices are , respectively. There are no repeated vertex labels when , i.e., .
Also in this labeling, there are two cases:
Case 1. When , i.e., .
In this case, we have will equal to when , so we change the labeling of two edges and as follows: and .
Then, , and .
Case 2. When , i.e., .
In this case, we have will equal to when , so we change the labeling of two edges () and () as follows: and .
Then, , and .
Note that does not follow this rule; however, it is an edge δ− graceful graph, as in Figure 11. In all cases, and are not congruent to nor which completes the proof.
Illustration 6. In Figure 12, we present with an edge 5− graceful labeling and with an edge 2− graceful labeling.
7. Edge δ− Graceful Labeling of the Gear Graph
Theorem 10. For any positive integer , the gear graph is an edge δ− graceful graph.
Proof. The gear graph is the graph obtained from the wheel by inserting a vertex between any two adjacent vertices in its cycle ; the gear graph has vertices and edges. Let the vertices of the wheel be , the hub vertex be , and the new vertices be , so the edges will be , see Figure 13.
We define the labeling function by
So the induced vertex labels will be
Hence, the labels of the vertices are , respectively, and the labels of the vertices are , respectively. Hence, there are no repeated vertex labels.
Case 1. When or . (i)If , then (ii)If , then
Case 2. When or . (i)If , then (ii)If , then In this case, we have will equal to , so we change the labeling of two edges and as follows: and . Then, and .
Case 3. When or .
(i)If , then (ii)If , then In this case, we have will equal to , so we change the labeling of two edges and as follows: and Then, , , , and
In all cases, is not congruent to nor , , and there are no repetition in the vertex labels which completes the proof.
Illustration 7. In Figure 14, we present with edge 7− graceful labeling, with edge 4− graceful labeling, and with edge 5− graceful labeling.
8. Edge δ− Graceful Labeling of the Closed Helm
The closed helm is the graph obtained from a helm by joining each pendent vertex to form a cycle; a closed helm has vertices and edges.
Theorem 11. For any positive integer , the closed helm is an edge δ− graceful graph.
Proof. In the closed helm , we have two copies of the cycle ; let the vertices in the inside copy be and the vertices on the outside copy be and a hub vertex ; the edges will be , see Figure 15.
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 will be , respectively, and the labels of the vertices will be , respectively.
Case 1. When or or or . (i)If , then (ii)If , then (iii)If , then (iv)If , then
Case 2. When . If , then
In this case, we have will equal to , so we change the labeling of two edges () and () as follows: and .
Then, , , , and .
Case 3. When . If , then
In this case, we have will equal to , so we change the labeling of two edges and as follows: and .
Then, , and .
Case 4. When . If , then .
In this case, we have will equal to , so we change the labeling of two edges () and () as follows: and .
Then, , , , and .
Case 5. When . If , then .
In this case, we have will equal to , so we change the labeling of two edges and as follows: and .
Then, and .
Clearly, in all cases, is not congruent to nor , , and all the labels of the vertices and are distinct and a multiplies of . Thus, the closed helm is an edge δ− graceful graph.
Illustration 8. In Figure 16, we present CH11 with edge 7− graceful labeling, CH12 with edge 6− graceful labeling, CH8 with edge 5− graceful labeling, CH14 with edge 3− graceful labeling, CH9 with edge 5− graceful labeling, and CH13 with edge 5− graceful labeling, respectively.
9. Edge δ− Graceful Labeling of the Friendship Frn
Theorem 12. For any positive integer , the friendship graph is an edge δ− graceful graph.
Proof. The friendship graph is a planar undirected graph constructed by joining copies of cycle graph with common vertex, so and . Let the friendship graph be given as in Figure 17 with central vertex , and the edges of will be . There are two cases:
Case 1. , or .
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 will be , , , , respectively, which are all distinct numbers. Also,
(i)If , then
(ii)If , then
Case 2. . We define the labeling function as follows: 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, which are all distinct numbers. (i)If , then Overall, all the labels of the vertices and are all distinct, multiple of , and different from which completes the proof.
Illustration 9. In Figure 18, we present the friendship graphs Fr9 with edge 5− graceful labeling and Fr10 with edge 6− graceful labeling.
10. Edge δ− Graceful Labeling of the Butterfly Graph
Theorem 13. For any positive integer , the butterfly graph is an edge δ− graceful graph.
Proof. The butterfly graph is a planer graph constructed by joining copies of cycle graph with a common edge, a butterfly graph has vertices and edges, let the common edge in all cycles be () and the vertices of one copy be and the vertices of the other copy be ; the edges will be {, }, see Figure 19. There are two cases:
Case 1. When is even. Define the labeling function as follows: In view of the above labeling pattern, then the induced vertex labels are In this labeling, we can check that Then
Case 2. When is odd. By using the same labeling as in the first case, in this case, we have will equal to , so we change the labeling of two edges () and () as follows: and , then , and .
We can see that the labels of the vertices , and , are all a multiple of and distinct numbers and different from the labels of the vertices and . Hence, the butterfly graph is an edge δ− graceful graph, for any positive integer .
Illustration 10. In Figure 20, we present with edge 5− graceful labeling and with edge 4− graceful labeling.
11. Conclusion
In the past few years, edge graceful labeling of graphs has been studied heavily and these topics continue to be attractive in the field of graph theory and discrete mathematics. A great number of published papers and results exist. So far, many graphs are unknown if it is edge graceful or not.
In this work, a new type of labeling called edge δ− graceful labeling is defined; the graph which satisfies the edge δ− graceful labeling is called an edge δ− graceful graph. Edge δ− graceful labeling of some special classes of graphs like wheel graph, double wheel, prism graph, gear graph, closed helm, friendship graph, and butterfly graph is investigated. In future work, we will study the necessary and sufficient conditions for some path-related graph to be an edge δ− graceful graph.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there is no conflict of interests.
Authors’ Contributions
The author read and approved the final manuscript.