Perfect matching and Hamilton cycle decomposition of complete balanced (k+1)-partite k-uniform hypergraphs

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Highlights

  • We generalize the perfect matching (Hamilton cycle) decomposition on complete graph and complete uniform hypergraph into complete (k+1)-parts k-uniform hypergraph Kk+1,n(k).

  • We prove our results using construction methods, which means that we can indeed find a perfect matching (Hamilton tight cycle) decomposition by computer in polynomial time.

  • Our construction methods will use some simple number theory knowledge.

Abstract

Let Kk+1,n(k) denote the complete balanced (k+1)-partite k-uniform hypergraph, whose vertex set consists of k+1 parts, each has n vertices and whose edge set contains all the k-element subsets with no two vertices from one part. In this paper, we prove that if kn and (nk,k)=1, then Kk+1,n(k) has a perfect matching decomposition; if (n,k)=1, then Kk+1,n(k) has a Hamilton tight cycle decomposition. In both cases, we use constructive methods which imply that we also give a polynomial algorithm to find a perfect matching decomposition or a Hamilton tight cycle decomposition.

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 V1,,V 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 K,n(k).

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 hyperedgesC=(v1,e1,v2,e2,,vn1,en1,vn,en,v1),where ei contains vi and vi+1(1in1), en contains vn and v1, {v1,,vn}=V(H), and e1,e2,,en 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 M (C) of H is a partitioning of all its edges such that each member of M (C) 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 K2n1 and K2nM (M is a perfect matching). The existence of Hamilton cycle decomposition for bipartite graphs K2n,2n and K2n1,2n1M (M is a perfect matching) was also proved in [11]. Obviously, the necessary condition for Knk to have a perfect matching decomposition is kn. Baranyai [2] proved that the necessary condition is also sufficient. In 1973 Bermond, Germa, Heydemann and Sotteau [5] made a conjecture that if n(nk), then the complete k-uniform hypergraph Knk on n vertices has a Hamilton Berge cycle decomposition. For k=3, 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 2kn2, there exists a Hamilton tight cycle decomposition of Knk if and only if n(nk). 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 Kk,n(k) has a perfect matching decomposition. The existence of a Hamilton tight cycle decomposition of Kk,n(k) was proved by Schroeder [12].

In this paper we consider the perfect matching and Hamilton tight cycle decomposition in complete (k+1)-partite k-uniform hypergraph Kk+1,n(k). Since the perfect matching covers all the vertices, if Kk+1,n(k) has a perfect matching, then we obtain that k(k+1)n, which implies that kn. We have the following two conjectures.

Conjecture 1

Let k ≥ 2. If kn, then there exists a perfect matching decomposition of Kk+1,n(k).

Conjecture 2

Let k ≥ 2. There exists a Hamilton tight cycle decomposition of Kk+1,n(k).

For these two conjectures, we have partial results in the paper.

Theorem 3

If kn and (nk,k)=1, then Kk+1,n(k) has a perfect matching decomposition.

Theorem 4

If (n,k)=1, then Kk+1,n(k) has a Hamilton tight cycle decomposition.

Section snippets

Some preliminaries

Given k ≥ 2, n ≥ 2, let Kk+1,n(k) be the complete balanced (k+1)-partite k-uniform hypergraph. Then |E(Kk+1,n(k))|=(k+1)nk. Let V1,,Vk+1 be the k+1 partite sets of V(Kk+1,n(k)) with |Vi|=n. Denote the elements of Vi by u1i=0,u2i=1,,uni=n1, where i{1,,k+1}. Given an edge {uj1,uj2,,ujk}E(Kk+1,n(k)), where ujiVji, we say that it is of type Vj1Vj2Vjk. Clearly, there are k+1 types of edges in Kk+1,n(k): V1V1V+1Vk+1, [k+1].

Let A be an n×(k+1) matrix. We call A a quasi-Latin square, if

Proof of Theorem 3

Given a matrix A=(ui,j) with AA. We divide the elements of A into the following (k+1)nk parts:{u1,1,u1,2,,u1,k},{u1,k+1,u2,1,,u2,k1},{u2,k,u2,k+1,,u3,k2},,{un,2,,un,k+1}.Denote Vi={u1,i,u2,i,,un,i}, 1ik+1. Then each part in (1) corresponds to an edge in Kk+1,n(k). In fact, these (k+1)nk parts in (1) make up a perfect matching, denote by MA, of Kk+1,n(k).

From the construction (1), the perfect matching MA in Kk+1,n(k) contains n/k edges being of type V2Vk+1 as follows:{uik,2,,uik,k+1}

Proof of Theorem 4

Given a matrix A=(ui,j) with AA. We construct a Hamilton tight cycle of Kk+1,n(k) as follows:CA=(u1,1,u1,2,,u1,k+1,u2,1,u2,2,,u2,k+1,,un,1,un,2,,un,k+1,u1,1),where any k consecutive vertices in the circular permutation form an edge. From the construction (4), it is easy to find that all the k+1 types of edges appear in CA circularly, whose period is k+1. Also CA contains n edges being of type V2Vk+1 as follows:{ui,2,,ui,k+1},i[n];and n edges being of type V1V1V+1Vk+1, 2k+1 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).

References (13)

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Cited by (4)

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).

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