A Novel Approach for Cyclic Decompositions of Balanced Complete Bipartite Graphs into Infinite Graph Classes

El-Mesady, A.; Bazighifan, Omar; Askar, S. S.

doi:https://doi.org/10.1155/2022/9308708

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Research Article | Open Access

Volume 2022 | Article ID 9308708 | https://doi.org/10.1155/2022/9308708

A Novel Approach for Cyclic Decompositions of Balanced Complete Bipartite Graphs into Infinite Graph Classes

A. El-Mesady,¹Omar Bazighifan,²and S. S. Askar³

Academic Editor: Serena Matucci

Received16 Feb 2022

Accepted08 Apr 2022

Published04 May 2022

Abstract

Graph theory is considered an attractive field for finding the proof techniques in discrete mathematics. The results of graph theory have applications in many areas of social, computing, and natural sciences. Graph labelings and decompositions have received much attention in the literature. Several types of graph labeling were proposed for solving the problem of decomposing different graph classes. In the present paper, we propose a technique for labeling the vertices of a bipartite graph with edges, called orthogonal labeling, to yield cyclic decompositions of balanced complete bipartite graphs by the graph . By applying the proposed orthogonal labeling technique, we had constructed decompositions of by paths, trees, one factorization, disjoint union of cycles, complete bipartite graphs, disjoint union of trees, caterpillars, and so forth. From the constructed results, we can confirm that the proposed orthogonal labeling technique is effective.

1. Introduction

To begin with, we summarize the main nomenclatures that will be used in this paper, which are listed in Table 1.

All graphs handled in this paper are simple (no loops or multiple edges). We refer the reader to [1] for any undefined terms. A graph labeling is a mapping that usually carries vertices and/or edges of a graph into a set of integers. Several kinds of labeling have been investigated; there is an excellent survey of graph labeling [2]. Sedlacek introduced the magic labeling of graphs [3], and the magic total labeling for the graph vertices first showed up in [4]. The super vertex magic total labeling notion was introduced by MacDougall et al. [5]. Emonoto et al. presented the notion of super edge-magic graphs [6]. Some results, concerned with super edge-magic total graphs, were introduced in [7]. The concepts of a V-super vertex magic labeling were handled by several authors [8, 9]. General results were presented in [10].

Definition 1. A graph is a subgraph of a graph ifand

Definition 2. Two simple graphs and are isomorphic () if there is a bijection such that if and only if .

Graph decompositions have been and remain the focus of a great deal of research [11]. A decomposition of any graph by a graph is a collection of isomorphic subgraphs called a -decomposition of where is a partition of the edge set of In [12], Alspach asked about the possibility of having a cycle decomposition of complete graphs. Several papers were devoted to solving this problem [13–15]. A graph is bipartite iff is composed of two disjoint independent sets of vertices called partite sets of Several kinds of graph decompositions appeared in the literature of graph theory. Among them, graph decompositions by bipartite graphs [16–19] were studied. Graph decompositions have interesting and many applications in different areas of computer science and combinatorics. Decompositions of complete bipartite graphs by cycles have also been studied in the literature [20–24]. Bipartite graphs can be used for modeling many systems. An increasing attention has been paid to such graphs, such as the disease analysis [25], controllability [26, 27], personal recommendation [28], link prediction [29], and community detection [30]. Factorizations of cyclic type for complete bipartite graphs into hypercubes were introduced in [31]. Decompositions of complete and complete bipartite graphs into cubes were proved in [32]. Cichacz et al. introduced the decomposition of complete bipartite graphs by generalized prisms [33]. Balanced complete bipartite graphs were decomposed by suns and stars in [34]. Hamilton cycle and path decompositions of complete bipartite graphs were investigated in [35]. In [36], the authors handled the decomposition of complete bipartite graphs by cycles of distinct even lengths. In [37], the authors proposed the decomposition of bipartite circulant graphs using algorithmic approaches. There are new applications for the decompositions of balanced complete bipartite graphs that appeared in [38, 39].

Motivated by the abovementioned results, based on the orthogonal labeling technique, we can construct several edge decompositions of balanced complete bipartite graphs by infinite graph classes. In order to show the effectiveness of the orthogonal labeling technique, we find several decompositions by paths, trees, cycles, complete bipartite graphs, caterpillars, and so forth. Our study is organized as follows. Section 2 addresses the proposed approach of orthogonal labeling. We introduce decompositions of balanced complete bipartite graphs by infinite graph classes in Section 3. Section 4 gives decompositions of balanced complete bipartite graphs by caterpillars. Finally, in Section 5, we introduce the conclusion and future works.

2. Proposed Approach

In what follows, we will show the proposed approach for constructing the edge decomposition of balanced complete bipartite graphs. Consider now the balanced complete bipartite graph with vertex set where and are two independent sets of vertices. There is a bijective mapping where the vertices in are labeled by and the vertices in are labeled by The distance between two vertices and is the usual circular distance defined by The edge is said to have length Given a subgraph with vertices and edges (isolated vertices are permitted), a labeling is an orthogonal labeling of if, for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even.

Example 1. An orthogonal labeling of the path is obtained if we label the vertices consecutively with Indeed, the edges and have length 1, the edge has length 0, and the edge has length 2. Similarly, we obtain an orthogonal labeling of the path by labeling the vertices consecutively with .

Definition 3. Let be a subgraph of Then, the subgraph with is called the-translate of

The next theorem relates the edge decomposition of complete bipartite graphs and orthogonal labeling.

Theorem 4. An edge decomposition of by a subgraph exists if and only if there exists an orthogonal labeling of .

Proof. We want to prove that for all By the way of contradiction, we assume that for with For the lengths which are repeated twice in let and be two edges of with length then and are different edges in having length But this is a contradiction, since has an orthogonal labeling. For the lengths 0 and (when is even) which are found once in let be an edge of with length then and are different edges in both having length But this is a contradiction, since has an orthogonal labeling.

Example 2. An edge decomposition of by is shown in Figure 1.

3. Decompositions of by Infinite Graph Classes

This section is devoted to prove the existence of several edge decompositions of which are based on the orthogonal labeling notion of different graphs.

Theorem 5. Let be an even integer, then there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 2. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once. Consequently, there is an edge decomposition of by .

Theorem 6. Let be an odd integer, then there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 3. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once. Consequently, there is an edge decomposition of by .

Theorem 7. Let then there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 4. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once. Consequently, there is an edge decomposition of by .

Theorem 8. Let be an integer, then there is an edge decomposition of by where .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once. Consequently, there is an edge decomposition of by .

Example 3. An orthogonal labeling of the path is shown in Figure 5.

Theorem 9. Let be an integer, then there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 6. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once. Consequently, there is an edge decomposition of by .

Theorem 10. Let be integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 7. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Consequently, there is an edge decomposition of by .

Theorem 11. Let be an odd integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 8. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once. Consequently, there is an edge decomposition of by .

Theorem 12. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is
see Figure 9. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Consequently, there is an edge decomposition of by .

Theorem 13. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 10. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 14. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 11. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once. Consequently, there is an edge decomposition of by .

Theorem 15. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 12. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Consequently, there is an edge decomposition of by .

Theorem 16. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by
and the edge set of is see Figure 13. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once. Consequently, there is an edge decomposition of by .

Theorem 17. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 14. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, the length is only present once. Hence, there is an edge decomposition of by .

4. Decompositions of by Caterpillars

Definition 18. The caterpillar graph is a tree obtained from the path by joining vertex to new vertices where are positive integers, and for .

Theorem 19. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 15. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 20. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 16. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Consequently, there is an edge decomposition of by .

Theorem 21. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 17. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 22. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 18. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 23. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 19. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 24. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 20. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 25. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 21. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 26. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 22. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

Theorem 27. Let be an integer such that Then, there is an edge decomposition of by .

Proof. Let the vertex set of be An orthogonal labeling of the subgraph can be defined by the mapping which is defined by and the edge set of is see Figure 23. It is straightforward to check that for every has precisely a pair of edges of length the length 0 is only present once, and the length is only present once if is even. Hence, there is an edge decomposition of by .

5. Conclusion

In this paper, we have successfully applied the orthogonal labeling notion to construct the decomposition of balanced complete bipartite graphs by infinite graph classes, such as paths, cycles, one factorization, complete bipartite graphs, disjoint union of trees, caterpillars, and so forth. From the constructed results, we can confirm that the proposed orthogonal labeling technique is effective. Here, we have concentrated on the orthogonal labeling of bipartite graphs. In future works, we will try to generalize the orthogonal labeling notion to other several classes of graphs.

Data Availability

The data used to support the findings of this study are available from the corresponding author on request.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

This work is supported by research supporting project number (RSP-2022/167), King Saud University, Riyadh, Saudi Arabia.

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