Abstract
In this article, we prove the existence of a simple cyclic near-resolvable -cycle system of for by the method of constructing its starter. Then, some new properties and results related to this construction are formulated.
1. Introduction
Throughout this paper, all graphs are considered undirected with vertices in where is odd. As usual, will denote the complete graph of order , and will denote the complete multigraph of order and multiplicity in which every two vertices are joined by parallel edges.
A -cycle system of a graph is a multiset of -cycles of whose edge sets partition . is said to be cyclic if and for each -cycle in we have that is also in , and it is said to be simple if all its cycles are distinct. A starter of cyclic -cycle system of is a multiset of -cycles that generates the multiset by repeated addition of 1 modulo . A near--factor of is a spanning ‐regular subgraph of for some vertex in .
A -cycle system of is said to be near-resolvable if its cycles can be partitioned into near--factors and is denoted by -. In general, it has been shown that there exists a near-resolvable -cycle system of if and only if is even and [1]. Such a near-resolvable -cycle system is cyclic if it is possible to label the vertices of with the elements of the cyclic group in such a way that for , where denotes the near--factor of obtained from by adding modulo to all its vertices. The near--factor is called a starter of cyclic near-resolvable -cycle system of .
The existence problem of -cycle systems of the complete multigraph has received much attention in recent years; this existence problem has been completely solved by Alspach and Gavlas [2] and by Šajna [3] for the important case when , and by Alspach et al. [4] for the case . An easier proof of the existence of odd cycle systems of using the difference method has been reproved by Buratti [5]. Then, Wu and Buratti [6] provided an algorithm to construct an explicit odd -cycle system of whenever it exists. In particular, the existence of cyclic -cycle systems of has been solved when [7, 8], [9], is even with [10], is a prime with the exception of [7], or is twice a prime power [11], and is thrice a prime [12]. Further results on cycle systems are in the surveys [13–15].
The necessary and sufficient conditions for the existence of cyclic -cycle system of and for the existence of simple cyclic -cycle system of , where is a prime, have been proved by Buratti et al. [16]. For odd, the necessary and sufficient conditions for decomposing into -cycles, and into cycles of prime length have been established by Smith in [17]. Shortly later, Bryant et al. [18] proved that the necessary and sufficient conditions for the existence of a -cycle system of for all are that is even and divides the number of edges in . More general results such as the existence problem for decomposing into cycles of varying lengths have been presented in [19, 20].
The problem of constructing near-resolvable -cycle system of has been contributed by many authors. A near-resolvable -cycle system of has been constructed for with except possibly values and except (for which such a system does not exist) [21], with or [22], with [23]. Recently, the existence of a near-resolvable -cycle system of for all and except possibly for and has been proved by Wang and Cao [24]. Previously, it has been proved that there exists a - for all odd and all [25]. In 2018, Matsubara and Kageyama [26] proved that a cyclic - exists if and only if .
In Section 2, we review some well-known definitions and preliminary results. Some introductory results are formulated in Section 3. Then, in Section 4, we explicitly construct a simple cyclic - for the case using a difference method. Moreover, we formulate some properties which are related to this construction. Finally, Section 5 discusses the conclusions and future work.
2. Preliminaries
In this section, we recall briefly some definitions and preliminary results that we used in the sequel. We start with the following definitions.
Definition 1 (see [27]). A path cover of a graph is a collection of vertex-disjoint paths of that covers the vertex set of .
Definition 2 (see [5]). Let be a graph and be an edge in . The difference of an edge is defined as .
Definition 3 (see [5, 28]). Let be a graph. The multiset is called the list of differences from . More generally, for a set of graphs, the list of differences from is the multiset which is obtained by linking together the ’s.
Definition 4 (see [6]). Let be a -cycle in . A cycle orbit of , denoted as , is a set of distinct -cycles in . A cycle orbit of is called full if its cardinality is ; otherwise, the cycle orbit of is short.
For convenience, we say is a full (short) cycle.
Definition 5 (see [5, 6]). Let be a -cycle in . The type of is the cardinality of the set .
From the above definition, it is obvious that if a cycle is of type 1 , then is a full (short) cycle.
Lemma 6 (see [5]). If is a -cycle in , then the type of is a common divisor of and .
The following lemma is a consequence of the theory developed in [16]. It will be crucial for proving our main results.
Lemma 7. Let be a multiset of -cycles of . Then, is a starter of cyclic -cycle system of if and only if covers exactly times.
3. Introductory Results
In this section, we introduce some definitions, notations, and introductory results required to establish our main results in the next section. We begin with defining relative path, relative cycle, and alternating arithmetic path that will be the basis for constructing the starter of simple cyclic near-resolvable -cycle system of .
Definition 8. Let be a graph of order , be a -path of , and be a -cycle of .(1)The -path is called the relative path of .(2)The -cycle is called the relative cycle of .
Lemma 9. Let be a graph of order .(1)If is a -path of and is the relative path of , then .(2)If is a -cycle of and is the relative cycle of , then .
Proof. (1)Suppose and are -path of and its relative path, respectively. The list of differences from and can be defined as Since is the relative path of , then , for all . Hence, substituting into (3), we obtain(2)The proof is similar to part .
Lemma 10. Let be a graph of order . If is a -cycle of and is the relative cycle of ; then, .
Proof. Let be a -cycle of and let be the relative cycle of . Assume on the contrary that ; then, there exists an integer such that . This implies thatSince is the relative cycle of , thenSolving (5) and (6) for and yieldsThis contradicts the fact that and are actually -cycles. Thus, and must have different orbits, so .
An alternating arithmetic path is a path with two sets of vertices satisfying certain conditions, as defined below.
Definition 11. Let and be positive integers with . An -alternating arithmetic path, denoted by , is a path of length with vertex set and edge set , such that the following properties are satisfied:(1) is constant, for all .(2) is constant, for all .
Definition 12. Let be an -alternating arithmetic path. The list of differences from is the multisetAccording to Definition 11, the -alternating arithmetic path either has odd order when or has even order when . Throughout, we use the following notations for -alternating arithmetic path of odd order and even order, respectively:In the following, we define a modulo scalar multiplication on paths and cycles in a finite graph of order , and then we prove some lemmas that will be used later in order to investigate some properties related to our construction.
Definition 13. Let , , and be positive integers with and . Let be a graph of order , be a -path of , and be a -cycle of .(1)The modulo multiplication of and is the -path .(2)The modulo multiplication of and is the -cycle .
Lemma 14. Let be a graph of order and be a -cycle of . If is any integer such that and , then(1).(2).
Proof. (1)Suppose that is a -cycle of . Then, Since is divisible by , then . Hence, (2)From the definition of modulo multiplication of and , we obtain But is divisible by , and this implies that . Hence,
Lemma 15. Let and be integers with and . Then, covers .
Proof. Let with . Assume on the contrary that . Then, we get and for some integers and .
Subtracting the above equations, we obtain . This implies Since , then and then from (14) we get . This implies that and therefore .
On the other hand, since and , then from (14) it follows that is a noninteger rational number. This contradicts the fact that is an integer. Thus, there are no such that , so covers
Lemma 16. Let be an integer; then, and are relatively prime.
Proof. Let be an integer such that divides both and . Then, there exists such thatFrom the equations above, we obtain . This implies that , and thenSince , then either or . Therefore, is the only positive integer which divides both and .
Now, we define a way of writing the cycle as linked vertex-disjoint paths. This way will be used mainly to prove the existence results in the following section.
Definition 17. Let be a -cycle, be a positive integer, and be a path cover of . The set of edges in that links the end of with the start of , for all where , is called the link set of .
Remark 18. Let be a -cycle, be a path cover of , and be a link set of . The cycle can be expressed as linked vertex-disjoint paths as follows:
Lemma 19. Let be a -cycle, be a path cover of , and be a link set of . Then, we have .
Proof. Let be the set of vertices of and the set of edges of . Based on Definition 3, the list of differences from is defined as a multiset consisting of the difference for each edge in as follows:Since is a path cover of , then Also, from the definition of link set of , we obtain Substituting (20) and (21) into (19) yields
To close this section, we provide an example below to demonstrate the concepts discussed in this section.
Example 20. Let , be a -cycle of . Then, the cycle can be written as linked vertex-disjoint paths as follows:where and are -alternating arithmetic paths and and are trivial paths. In addition, the set of four edges that links the paths and , respectively, along the cycle is considered the link set for the path cover .
Based on Definition 8, the relative cycle of is . It is easy to see that the sum of each pair of corresponding vertices of and its relative cycle is equal to (the order of .
Since , thenIn other words, as shown in part of Lemma 14.
4. Simple Cyclic Near-Resolvable -Cycle System of
In this section, we prove, explicitly and directly, the existence of a simple cyclic near-resolvable -cycle system of by constructing its starter.
To construct a simple cyclic near-resolvable -cycle system of , it is enough to exhibit a starter of cyclic -cycle system of which satisfies a near-2-factor and contains no two cycles in the same orbit. Let us provide an example to illustrate the above definition.
Example 21. Let and be a set of -cycles of such that and .
Easily, it can be observed that the -cycles of are vertex-disjoint and cover each nonzero element of exactly once. Hence, we can say that is a -regular graph satisfying the near-2-factor with focus zero.
In order to show that is a starter of cyclic -cycle system of , we need to calculate the list of differences from as illustrated in Table 1.
As listed in Table 1, each nonzero element of occurs twice in . Then, by Lemma 7, is a starter set of cyclic -cycle system of .
Since the sum of each pair of corresponding vertices of and is equal to 9 the order of , then is the relative cycle of , and so, by Lemma 10, . From Definition 4, we conclude that all the generated cycles by repeated addition of 1 modulo to contain no repetitions.
Now, satisfies all the conditions to be a starter of simple cyclic near-resolvable -cycle system of . Once the starter set has been provided, all cycles of simple cyclic - can be generated by repeated addition of 1 modulo 9 as shown in Table 2.
In the following, we construct a simple cyclic near-resolvable -cycle system of . Since the construction is different depending on whether is odd or even, we classify the construction into two cases: when is odd and when is even.
Lemma 22. For any positive odd integer , there exists a simple cyclic near-resolvable -cycle system of .
Proof. Let , where is a positive odd integer. Let and be the -cycles of defined as linked vertex-disjoint paths as follows:whereSince is a positive odd integer, then any -alternating arithmetic path and -alternating arithmetic path have even order. As illustrated in Figure 1, the construction of and can be described in terms of their vertices as for .
In this way, we note that in the cycle the ’s with odd and the ’s with even form the following increasing sequences, respectively:
in the interval and
in the complement of in .
In contrast, in the ’s with odd and the ’s with even form the following decreasing sequences, respectively:
in and
in the complement of in .
Thus, for , the vertices in are pairwise distinct and hence is actually -cycle.
In the rest of this proof, three parts are considered to prove that the set of cycles satisfies the conditions to be a starter of simple cyclic near-resolvable -cycle system of .
Part 1. In this part, we prove that satisfies the near-two-factor condition. This will be verified by proving that the union of vertex sets of and covers each element of exactly once. The vertex sets of and can be enumerated by the union of vertex sets of all linked paths in both and , respectively.whereAccording to the above vertex sets, it can be easily noted that each nonzero element of occurs exactly once in . Since any cycle is a -regular graph and , then the set of cycles forms near-two-factor with focus zero.
Part 2. This part shows that the set of cycles is a starter of cyclic -cycle system of . For this part, it is sufficient to prove that the list of differences from covers exactly twice.
Based on Definition 3, the list of differences from is defined as . Then, from Lemma 19 and Definition 12, the list of differences from iswhereAs shown above, each nonzero element of appears exactly once in .
From (25), we can deduce that is the relative cycle of . Hence, by part of Lemma 9, we obtain . Now, we conclude that each nonzero element of appears exactly twice in . Based on Lemma 7, the set of cycles is a starter of cyclic -cycle system of for all odd positive integer .
Part 3. We check that all the generated cycles from the starter contain no repetitions by showing that all the cycles of have different orbit.
Since is the relative cycle of , then by Lemma 10, . Thus, all the generated cycles by repeated addition of 1 modulo to contain no repetitions.
By summing up the former three parts, we have proved that, for any positive odd integer , the set of cycles is a starter of simple cyclic near-resolvable -cycle system of .
Lemma 23. For any nonnegative even integer , there exists a simple cyclic near-resolvable -cycle system of .
Proof. Let , where is a nonnegative even integer. Let and be the -cycles of defined as linked vertex-disjoint paths as follows:whereSince is a nonnegative even integer, then any -alternating arithmetic path and -alternating arithmetic path have odd order. As shown in Figure 2, the construction of and can be described in terms of their vertices as for .
The rest of this proof is similar to the proof of Lemma 22, hence omitted.
Theorem 24. For each positive , there exists a simple cyclic near-resolvable -cycle system of .
Proof. The proof is immediate from Lemmas 22 and 23.
By reviewing the construction of a starter of simple cyclic near-resolvable -cycle system of , as shown in Figures 1 and 2, the construction has a butterfly shape in which each cycle represents a side of symmetrical butterfly wings. If given one cycle of the starter set, the other is the relative cycle of .
Next, some related properties of the starter of simple cyclic near-resolvable -cycle system of will be formulated.
Lemma 25. Let , , and be integers such that and . If is a starter of simple cyclic near-resolvable -cycle system of , then covers .
Proof. Since is a starter of simple cyclic near-resolvable -cycle system of , then satisfies the near-two-factor with focus zero. This implies that . Then, by Lemma 15, we obtain the notion that covers .
Theorem 26. Let , , and be integers with and . If is a starter of simple cyclic near-resolvable -cycle system of that satisfies (25) or (32), then and are the same starter of simple cyclic near-resolvable -cycle system of .
Proof. Suppose that is a starter of simple cyclic near-resolvable -cycle system of that satisfies (25) or (32). Then, is the relative cycle of and hence, by part and of Lemma 14, we have To prove that is a starter of simple cyclic near-resolvable -cycle system of , we need to prove the following:(1) covers .(2) covers exactly twice. The first condition is satisfied from Lemma 25. Based on the definition of , for each edge in we have that is an edge in .
Suppose is the difference of the edge for , where . Then, the difference of the edge is , where and .
Since is a starter of simple cyclic near-resolvable -cycle system of , then covers exactly twice. But is the relative cycle of ; this implies that covers exactly once. Therefore, also covers exactly once (from Lemma 15); thus, covers exactly twice.
In the following example, we construct some of distinct starters of simple cyclic - in accordance with the theorem above.
Example 27. Let be an integer with and , with vertices in and be a set of -cycles of where and .
In Example 21, it is proved that is a starter of simple cyclic -. The next table shows the possible values of that make the set of cycles be a starter of simple cyclic -.
In Table 3, it can be remarked that each set of cycles covers exactly once and the list of differences from covers exactly twice. Therefore, we conclude that for any such that both sets and are the same starter of simple cyclic -.
The simple cyclic near-resolvable -cycle system of is a array if its starter is a set of full cycles. By the following theorem, we can answer the question whether the starter of simple cyclic near-resolvable -cycle system of is set of full cycles or not.
Theorem 28. If is a nonnegative integer, then any starter of simple cyclic near-resolvable -cycle system of is a set of full cycles.
Proof. Let and be -cycles in . Suppose that is a starter of simple cyclic near-resolvable -cycle system of .
From Lemma 16, we have that and are relatively prime, which implies that the only positive common divisor of and is 1. By Lemma 6, since the type of any -cycle in is a common divisor of and , it follows that any cycle in is of type . Consequently, any starter of simple cyclic near-resolvable -cycle system of is a set of full cycles.
5. Conclusions
This article has proposed near-resolvable -cycle system of as an edge-decomposition of the complete multigraph into classes of -cycles such that each class satisfies the near-2-factor. In particular, the difference method has been exploited to construct a simple cyclic near-resolvable -cycle system of for the odd case , and this construction has been exemplified for the case . Finally, we have formulated some properties of this construction. We expect that this study can be developed and extended to construct a simple cyclic near-resolvable -cycle system of for the case odd.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors are very grateful to the anonymous referees for their valuable comments and suggestions. This work is partially financially supported by the Universiti Utara Malaysia through the Fundamental Research Grant Scheme (FRGS) [Code: 14198].