Perfect matching and Hamilton cycle decomposition of complete balanced -partite k-uniform hypergraphs
Introduction
A k-uniform hypergraph H is a pair (V, E), where V is a finite set of vertices and E is a family of k-element subsets of V. If the vertex set V can be partitioned into ℓ parts with n vertices in each part, and the edge set E contains all the k-element subsets with no two vertices from one part, we denote the k-uniform hypergraph by .
A matching of size s in H is a family of s pairwise disjoint edges of H. If the matching covers all the vertices of H, then we call it a perfect matching. There are several notions of Hamilton cycle in a hypergraph. Berge [3] generalized the definition of a Hamilton cycle C′ of H as a sequence of vertices and hyperedgeswhere ei contains vi and en contains vn and v1, and are distinct. Kierstead and Katona [8] introduced a more structured definition of Hamilton cycle: A cyclic ordering of the vertices of a k-uniform hypergraph is called a Hamiltonian cycle if any k consecutive vertices in the ordering form an edge. A Hamilton Kierstead Katona cycle is often called a Hamilton tight cycle. Clearly, a Hamilton tight cycle is also a Hamilton Berge cycle, but not conversely.
A perfect matching (resp. Hamilton cycle) decomposition () of H is a partitioning of all its edges such that each member of () is a perfect matching (resp. Hamilton cycle). It is well-known that K2n and Kn,n always have a perfect matching decomposition. Walecki [10] showed that there is a Hamilton cycle decomposition for and (M is a perfect matching). The existence of Hamilton cycle decomposition for bipartite graphs K2n,2n and (M is a perfect matching) was also proved in [11]. Obviously, the necessary condition for to have a perfect matching decomposition is k∣n. Baranyai [2] proved that the necessary condition is also sufficient. In 1973 Bermond, Germa, Heydemann and Sotteau [5] made a conjecture that if then the complete k-uniform hypergraph on n vertices has a Hamilton Berge cycle decomposition. For this conjecture follows by the results of Bermond [6] and Verrall [13]. Kühn and Osthus [7] showed when 4 ≤ k < n and n ≥ 30, the conjecture holds as well. In 2010 Bailey and Stevens [1] conjectured that when n ≥ 5 and there exists a Hamilton tight cycle decomposition of if and only if . Meszka and Rosa [9] along with Bailey and Stevens [1] showed that the conjecture holds for some special values of n and k. By Theorem 3 of [4], the complete k-partite k-uniform hypergraph has a perfect matching decomposition. The existence of a Hamilton tight cycle decomposition of was proved by Schroeder [12].
In this paper we consider the perfect matching and Hamilton tight cycle decomposition in complete -partite k-uniform hypergraph . Since the perfect matching covers all the vertices, if has a perfect matching, then we obtain that which implies that k∣n. We have the following two conjectures. Conjecture 1 Let k ≥ 2. If k∣n, then there exists a perfect matching decomposition of . Conjecture 2 Let k ≥ 2. There exists a Hamilton tight cycle decomposition of .
For these two conjectures, we have partial results in the paper. Theorem 3 If k∣n and then has a perfect matching decomposition. Theorem 4 If then has a Hamilton tight cycle decomposition.
Section snippets
Some preliminaries
Given k ≥ 2, n ≥ 2, let be the complete balanced ()-partite k-uniform hypergraph. Then . Let be the partite sets of with . Denote the elements of Vi by where . Given an edge where we say that it is of type . Clearly, there are types of edges in : .
Let A be an matrix. We call A a quasi-Latin square, if
Proof of Theorem 3
Given a matrix with . We divide the elements of A into the following parts:Denote . Then each part in (1) corresponds to an edge in . In fact, these parts in (1) make up a perfect matching, denote by MA, of .
From the construction (1), the perfect matching MA in contains n/k edges being of type as follows:
Proof of Theorem 4
Given a matrix with . We construct a Hamilton tight cycle of as follows:where any k consecutive vertices in the circular permutation form an edge. From the construction (4), it is easy to find that all the types of edges appear in CA circularly, whose period is . Also CA contains n edges being of type as follows:and n edges being of type as
Acknowledgments
The authors are thankful to the anonymous referee for his/her useful comments. Yi Zhang is supported by the National Natural Science Foundation of China (Grant 11901048) and the China Postdoctoral Science Foundation (NO. 2019M660562). Mei Lu is supported by the National Natural Science Foundation of China (Grant 11771247).
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- 1
National Natural Science Foundation of China (Grant 11901048) and the China Postdoctoral Science Foundation (NO. 2019M660562).
- 2
National Natural Science Foundation of China (Grant 11771247).