FFACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS
M. A. BAHMANIAN
Abstract.
In a mathematics workshop with mn mathematicians from n different areas,each area consisting of m mathematicians, we want to create a collaboration network. Forthis purpose, we would like to schedule daily meetings between groups of size three, sothat (i) two people of the same area meet one person of another area, (ii) each person hasexactly r meeting(s) each day, and (iii) each pair of people of the same area have exactly λ meeting(s) with each person of another area by the end of the workshop. Using hypergraphamalgamation-detachment, we prove a more general theorem. In particular we show thatabove meetings can be scheduled if: 3 (cid:23) rm , 2 (cid:23) rnm and r (cid:23) λ p n ´ q ` m ˘ . This result canbe viewed as an analogue of Baranyai’s theorem on factorizations of complete multipartitehypergraphs. Introduction
Throughout this paper, N is the set of positive integers, m, n, r, λ P N , and r n s : “t , . . . , n u . In a mathematics workshop with mn mathematicians from n different areas,each area consisting of m mathematicians, we want to create a collaboration network. Forthis purpose, we would like to schedule daily meetings between groups of size three, so that(i) two people of the same area meet one person of another area, (ii) each person has ex-actly r meeting(s) each day, and (iii) each pair of people of the same area have exactly λ meeting(s) with each person of another area by the end of the workshop. Using hypergraphamalgamation-detachment, we prove a more general theorem. In particular we show thatabove meetings can be scheduled if: 3 (cid:23) rm , 2 (cid:23) rnm and r (cid:23) λ p n ´ q ` m ˘ .A hypergraph G is a pair p V, E q where V is a finite set called the vertex set, E is the edgemultiset, where every edge is itself a multi-subset of V . This means that not only can anedge occur multiple times in E , but also each vertex can have multiple occurrences withinan edge. The total number of occurrences of a vertex v among all edges of E is called the degree , d G p v q of v in G . For h P N , G is said to be h - uniform if | e | “ h for each e P E . For r, r , . . . , r k P N , an r -factor in a hypergraph G is a spanning r -regular sub-hypergraph, and Date : February 8, 2021.2000
Mathematics Subject Classification.
Key words and phrases.
Baranyai’s Theorem, Amalgamations, Detachments, Multipartite Hypergraphs, Fac-torizations, Decompositions.Research is partially supported by NFIG at ISU, and NSA Grant H98230-16-1-0304. a r X i v : . [ m a t h . C O ] F e b ACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS 2 an p r , . . . , r k q -factorization is a partition of the edge set of G into F , . . . , F k where F i is an r i -factor for i P r k s . We abbreviate p r, . . . , r q -factorization to r -factorization.The hypergraph K hn : “ p V, ` Vh ˘ q with | V | “ n (by ` Vh ˘ we mean the collection of all h -subsets of V ) is called a complete h -uniform hypergraph. In connection with Kirkman’sschoolgirl problem [14], Sylvester conjectured that K hn is 1-factorable if and only if h (cid:23) n . Thisconjecture was finally settled by Baranyai [8]. Let K n ˆ m denote the 3-uniform hypergraphwith vertex partition t V i : i P r n su , so that V i “ t x ij : j P r m su for i P r n s , and with edge set E “ tt x ij , x ij , x kl u : i, k P r n s , j, j , l P r m s , j ‰ j , i ‰ k u . One may notice that finding an r -factorization for K n ˆ m is equivalent to scheduling the meetings between mathematicianswith the above restrictions for the case λ “ e of G by λ copies of e , then we denote the new hypergraph by λ G . In this paper, the main result is the following theorem which is obtained by proving amore general result (see Theorem 3.1) using amalgamation-detachment techniques. Theorem 1.1. λ K m ˆ n is p r , . . . , r k q -factorable if (S1) 3 (cid:23) r i m for i P r k s , (S2) 2 (cid:23) r i mn for i P r k s , and (S3) ř ki “ r i “ λ p n ´ q ` m ˘ . In particular, by letting r “ r “ ¨ ¨ ¨ “ r k in Theorem 1.1, we solve the MathematiciansCollaboration Problem in the following case. Corollary 1.2. λ K m ˆ n is r -factorable if (i) 3 (cid:23) rm , (ii) 2 (cid:23) rnm , and (iii) r (cid:23) λ p n ´ q ` m ˘ . The two results above can be seen as analogues of Baranyai’s theorem for complete 3-uniform “multipartite” hypergraphs. We note that in fact, Baranyai [9] solved the problemof factorization of complete uniform multipartite hypergraphs, but here we aim to solve thisproblem under a different notion of “multipartite”. In Baranyai’s definition, an edge canhave at most one vertex from each part, but here we allow an edge to have two verticesfrom each part (see the definition of K m ˆ n above). More precise definitions together withpreliminaries are given in Section 2, the main result is proved in Section 3, and related openproblems are discussed in the last section.Amalgamation-detachment technique was first introduced by Hilton [10] (who found anew proof for decompositions of complete graphs into Hamiltonian cycles), and was moredeveloped by Hilton and Rodger [11]. Hilton’s method was later genealized to arbitrary ACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS 3 graphs [5], and later to hypergraphs [1, 2, 7, 4] leading to various extensions of Baranyai’stheorem (see for example [1, 3]). The results of the present paper, mainly relies on thosefrom [1] and [15]. For the sake of completeness, here we give a self contained exposition.2.
More Terminology and Preliminaries
Recall that an edge can have multiple copies of the same vertex. For the purpose ofthis paper, all hypergraphs (except when we use the term graph) are 3-uniform, so an edgeis always of one of the forms t u, u, u u , t u, u, v u , and t u, v, w u which we will abbreviate to t u u , t u , v u , and t u, v, w u , respectively. In a hypergraph G , mult G p . q denotes the multiplicity;for example mult G p u q is the multiplicity of an edge of the form t u u . Similarly, for a graph G , mult p u, v q is the multiplicity of the edge t u, v u . A k-edge-coloring of a hypergraph G isa mapping K : E p G q Ñ r k s , and the sub-hypergraph of G induced by color i is denoted by G p i q . Whenever it is not ambiguous, we drop the subscripts, and also we abbreviate d G p i q p u q to d i p u q , mult G p i q p u q to mult i p u q , etc..Factorizations of the complete graph, K n , is studied in a very general form in [12, 13],however for the purpose of this paper, a λ -fold version is needed: Theorem 2.1. (Bahmanian, Rodger [6, Theorem 2.3]) λK n is p r , . . . , r k q -factorable if andonly if r i n is even for i P r k s and ř ki “ r i “ λ p n ´ q . Let K ˚ n denote the 3-uniform hypergraph with n vertices in which mult p u , v q “
1, andmult p u q “ mult p u, v, w q “ u, v, w . A (3-uniform) hypergraph G “p V, E q is n -partite , if there exists a partition t V , . . . , V n u of V such that for every e P E , | e X V i | “ , | e X V j | “ i, j P r n s with i ‰ j . For example, both K ˚ n and K m ˆ n are n -partite. We need another simple but crucial lemma: Lemma 2.2. If r i n is even for i P r k s , and ř ki “ r i “ λ p n ´ q , then λK ˚ n is p r , . . . , r k q -factorable.Proof. Let G “ λK n with vertex set V . By Theorem 2.1, G is p r , . . . , r k q -factorable. Usingthis factorization, we obtain a k -edge-coloring for G such that d G p i q p v q “ r i for every v P V and every color i P r k s . Now we form a k -edge-colored hypergraph H with vertex set V suchthat mult H p i q p u , v q “ mult G p i q p u, v q for every pair of distinct vertices u, v P V , and eachcolor i P r k s . It is easy to see that H – λK ˚ n and d H p i q p v q “ r i for every v P V and everycolor i P r k s . Thus we obtain a p r , . . . , r k q -factorization for λK ˚ n . (cid:3) If the multiplicity of a vertex α in an edge e is p , we say that α is incident with p distinct hinges , say h p α, e q , . . . , h p p α, e q , and we also say that e is incident with h p α, e q , . . . , h p p α, e q . ACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS 4
The set of all hinges in G incident with α is denoted by H G p α q ; so | H G p α q| is in fact thedegree of α .Intuitively speaking, an α -detachment of a hypergraph G is a hypergraph obtained bysplitting a vertex α into one or more vertices and sharing the incident hinges and edgesamong the subvertices. That is, in an α -detachment G of G in which we split α into α and β , an edge of the form t α p , u , . . . , u z u in G will be of the form t α p ´ i , β i , u , . . . , u z u in G for some i , 0 ď i ď p . Note that a hypergraph and its detachments have the samehinges. Whenever it is not ambiguous, we use d , mult , etc. for degree, multiplicity andother hypergraph parameters in G .Let us fix a vertex α of a k -edge-colored hypergraph G “ p V, E q . For i P r k s , let H i p α q bethe set of hinges each of which is incident with both α and an edge of color i (so d i p α q “| H i p α q| ). For any edge e P E , let H e p α q be the collection of hinges incident with both α and e . Clearly, if e is of color i , then H e p α q Ă H i p α q .A family A of sets is laminar if, for every pair A, B of sets belonging to A , either A Ă B ,or B Ă A , or A X B “ ∅ . We shall present two lemmas, both of which follow immediatelyfrom definitions. Lemma 2.3.
Let A “ t H p α q , . . . , H k p α qu Y t H e p α q : e P E u . Then A is a laminar familyof subsets of H p α q . For each p P t , u , and each U Ă V zt α u , let H p α p , U q be the set of hinges each of which isincident with both α and an edge of the form t α p u Y U in G (so | H p α p , U q| “ p mult pt α p , U u ). Lemma 2.4.
Let B “ t H p α p , U q : p P t , u , U Ă V zt α uu . Then B is a laminar family ofdisjoint subsets of H p α q . If x, y are real numbers, then t x u and r x s denote the integers such that x ´ ă t x u ď x ď r x s ă x `
1, and x « y means t y u ď x ď r y s . We need the following powerful lemma: Lemma 2.5. (Nash-Williams [15, Lemma 2]) If A , B are two laminar families of subsetsof a finite set S , and n P N , then there exist a subset A of S such that | A X P | « | P |{ n for every P P A Y B . Proofs
Notice that λ K m ˆ n is a 3 λ p n ´ q ` m ˘ -regular hypergraph with nm vertices and 2 λm ` n ˘` m ˘ edges. To prove Theorem 1.1, we prove the following seemingly stronger result. Theorem 3.1.
Let (cid:23) r i m and (cid:23) r i mn for i P r k s , and ř ki “ r i “ λ p n ´ q ` m ˘ . Then forall (cid:96) “ n, n ` , . . . , mn there exists a k -edge-colored (cid:96) -vertex n -partite hypergraph G “ p V, E q and a function g : V Ñ N such that the following conditions are satisfied: ACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS 5 (C1) ř v P W g p v q “ m for each part W of G ; (C2) mult p u , v q “ λ ` g p u q ˘ g p v q for each pair of vertices u, v from different parts of G ; (C3) mult p u, v, w q “ λg p u q g p v q g p w q for each pair of distinct vertices u, w from the samepart, and v from a different part of G ; (C4) d i p u q “ r i g p u q for each color i P r k s and each u P V . Remark 3.2.
It is implicitly understood that every other type of edge in G is of multiplicity0. Before we prove Theorem 3.1, we show how Theorem 1.1 is implied by Theorem 3.1. Proof of Theorem 1.1.
It is enough to take (cid:96) “ mn in Theorem 3.1. Then there exists an n -partite hypergraph G “ p V, E q of order mn and a function g : V Ñ N such that by (C1) ř v P W g p v q “ m for each part W of G . This implies that g p v q “ v P V and thateach part of G has m vertices. By (C2), mult G p u , v q “ λ ` ˘ p q “ u, v from different parts of G , and by (C3), mult G p u, v, w q “ λ for each pair of vertices u, v from the same part and w from a different part of G . This implies that G – λ K m ˆ n . Finally,by (C4), G admits a k -edge-coloring such that d G p i q p v q “ r i for each color i P r k s . Thiscompletes the proof. (cid:3) The idea of the proof of Theorem 3.1 is that each vertex α will be split into g p α q verticesand that this will be done by “splitting off” single vertices one at a time. Proof of Theorem 3.1.
We prove the theorem by induction on (cid:96) .First we prove the basis of induction, case (cid:96) “ n . Let G “ p V, E q be λm ` m ˘ K ˚ n and let g p v q “ m for all v P V . Since G has n vertices, it is n -partite (each vertex being a partiteset). Obviously, ř v P W g p v q “ g p v q “ m for each part W of G . Also, mult p u , v q “ λm ` m ˘ “ λ ` g p u q ˘ g p v q for each pair of vertices u, v from distinct parts of G , so (C2) is satisfied. Sincethere is only one vertex in each part, (C3) is trivially satisfied.Since for i P r k s , 2 (cid:23) r i mn and ř ki “ r i m “ λm p n ´ q ` m ˘ , by Lemma 2.2, G is p mr , . . . , mr k q -factorable. Thus, we can find a k -edge-coloring for G such that d G p j q p v q “ mr i “ r i g p v q for i P r k s , and therefore (C4) is satisfied.Suppose now that for some (cid:96) P t n, n ` , . . . , mn ´ u , there exists a k -edge-colored n -partitehypergraph G “ p V, E q of order (cid:96) and a function g : V Ñ N satisfying properties (C1)–(C4)from the statement of the theorem. We shall now construct an n -partite hypergraph G oforder (cid:96) ` g : V p G q Ñ N satisfying (C1)–(C4).Since (cid:96) ă mn , G is n -partite and (C1) holds for G , there exists a vertex α of G with g p α q ą
1. The graph G will be constructed as an α -detachment of G with the help of ACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS 6 laminar families A : “ t H p α q , . . . , H k p α qu Y t H e p α q : e P E u and B : “ t H p α p , U q : p P t , u , U Ă V zt α uu . By Lemma 2.5, there exists a subset Z of H p α q such that(1) | Z X P | « | P |{ g p α q , for every P P A Y B . Let G “ p V , E q with V “ V Y t β u be the hypergraph obtained from G by splitting α into two vertices α and β in such a way that hinges which were incident with α in G becomeincident in G with α or β according to whether they do not or do belong to Z , respectively.More precisely,(2) H p β q “ Z, H p α q “ H p α qz Z. So G is an α -detachment of G and the colors of the edges are preserved. Let g : V Ñ N sothat g p α q “ g p α q ´ , g p β q “
1, and g p u q “ g p u q for each u P V zt α, β u . It is obvious that G is of order (cid:96) ` n -partite, and ř v P W g p v q “ m for each part W of G (the new vertex β belongs to the same part of G as α belongs to). Moreover, it is clear that G satisfies(C2)–(C4) if t α, β u X t u, v, w u “ H . For the rest of the argument, we will repeatedly usethe definitions of A , B , (1), and (2).For i P r k s we have d i p β q “ | Z X H i p α q| « | H i p α q|{ g p α q “ d i p α q{ g p α q “ r i “ r i g p β q ,d i p α q “ d i p α q ´ d i p β q “ r i g p α q ´ r i “ r i p g p α q ´ q “ r i g p α q , so G satisfies (C4).Let u P V so that u and α (or β ) belong to different parts of G . We havemult p β, u q “ | Z X H p α, t u uq| « | H p α, t u uq|{ g p α q “ mult p α, u q{ g p α q“ λ ˆ g p u q ˙ “ λ ˆ g p u q ˙ g p β q , mult p α, u q “ mult p α, u q ´ mult p β, u q “ λ ˆ g p u q ˙ g p α q ´ λ ˆ g p u q ˙ “ λ ˆ g p u q ˙ g p α q . Recall that g p α q ě
2, and for every e P E and i P r k s , | H e p α q| ď
2, and thus | Z X H e p α q| «| H e p α q|{ g p α q ď
1. This implies thatmult p β , u q “ “ λ ˆ g p β q ˙ g p u q , ACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS 7 and so mult p α , u q “ mult p α , u q ` mult p α, β, u q . Now we havemult p α, β, u q “ | Z X H p α , t u uq| « | H p α , t u uq|{ g p α q“ p α , u q{ g p α q “ λ p g p α q ´ q g p u q “ λg p α q g p β q g p u q , mult p α , u q “ mult p α , u q ´ mult p α, β, u q “ λ ˆ g p α q ˙ g p u q ´ λ p g p α q ´ q g p u q“ λ ˆ g p α q ´ ˙ g p u q “ λ ˆ g p α q ˙ g p u q . Therefore G satisfies (C2).Let u, v P V so that u, v belong to different parts of G , u, α belong to the same part of G , and u R t α, β u . We havemult p β, u, v q “ | Z X H p α, t u, v uq| « | H p α, t u, v uq|{ g p α q “ mult p α, u, v q{ g p α q“ λg p u q g p v q “ λg p β q g p u q g p v q , mult p α, u, v q “ mult p α, u, v q ´ mult p β, u, v q “ λ p g p α q ´ q g p u q g p v q “ λg p α q g p u q g p v q . Finally, let u, v P V so that u, v belong to the same part of G , and u, α belong to differentparts of G , and u R t α, β u . By an argument very similar to the one above, we havemult p u, v, β q “ λg p u q g p v q g p β q , mult p u, v, α q “ λg p u q g p v q g p α q . Therefore G satisfies (C3), and the proof is complete. (cid:3) Final Remarks
We define K m ,...,m n similar to K m ˆ n with the difference that in K m ,...,m n we allow differentparts to have different sizes. It seems reasonable to conjecture that Conjecture 4.1. λ K m ,...,m n is p r , . . . , r k q -factorable if and only if (i) m i “ m j : “ m for i, j P r n s , (ii) 3 (cid:23) r i mn for i P r k s , and (iii) ř ki “ r i “ λ p n ´ q ` m ˘ . We prove the necessity as follows. Since λ K m ˆ n is factorable, it must be regular. Let u and v be two vertices from two different parts, say p th and q th parts respectively. Then we ACTORIZATIONS OF COMPLETE MULTIPARTITE HYPERGRAPHS 8 have the following sequence of equivalences: d p u q “ d p v q ðñ ÿ ď i ď ni ‰ p ˆ m i ˙ ` p m p ´ q ÿ ď i ď ni ‰ p m i “ ÿ ď i ď ni ‰ q ˆ m i ˙ ` p m q ´ q ÿ ď i ď ni ‰ q m i ðñ ˆ m q ˙ ` ÿ ď i ď ni ‰ p,q ˆ m i ˙ ` p m p ´ qp m q ` ÿ ď i ď ni ‰ p,q m i q “ ˆ m p ˙ ` ÿ ď i ď ni ‰ p,q ˆ m i ˙ ` p m q ´ qp m p ` ÿ ď i ď ni ‰ p,q m i q ðñ ˆ m p ˙ ´ ˆ m q ˙ ` m p m q ´ m p ´ m p m q ` m q ` p m p ´ m q q ÿ ď i ď ni ‰ p,q m i q “ ðñ m p ´ m q ´ m p ` m q ` p m p ´ m q q ÿ ď i ď ni ‰ p,q m i q “ ðñp m p ´ m q qp m p ` m q ´ ` ÿ ď i ď ni ‰ p,q m i q “ ðñ m p “ m q : “ m. This proves (i). The existence of an r i -factor implies that 3 (cid:23) r i mn for i P r k s . Since each r i -factor is an r i -regular spanning sub-hypergraph and λ K m ˆ n is 3 λ p n ´ q ` m ˘ -regular, wemust have ř ki “ r i “ λ p n ´ q ` m ˘ .In Theorem 1.1, we made partial progress toward settling Conjecture 4.1, however at thispoint, it is not clear to us whether our approach will work for the remaining cases.5. Acknowledgement
The author is deeply grateful to Professors Chris Rodger, Mateja ˇSajna, and the anony-mous referee for their constructive comments.
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