Factorizations of Complete Multipartite Hypergraphs

02/04/2021 ∙ by Amin Bahmanian, et al. ∙ 0

In a mathematics workshop with mn mathematicians from n different areas, each area consisting of m mathematicians, we want to create a collaboration network. For this 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 exactly 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 hypergraph amalgamation-detachment, we prove a more general theorem. In particular we show that above meetings can be scheduled if: 3 | rm, 2 | rnm and r | 3λ(n-1)m2. This result can be viewed as an analogue of Baranyai's theorem on factorizations of complete multipartite hypergraphs.

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1. Introduction

Throughout this paper, is the set of positive integers, , and . In a mathematics workshop with mathematicians from different areas, each area consisting of mathematicians, we want to create a collaboration network. For this 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 exactly meeting(s) each day, and (iii) each pair of people of the same area have exactly meeting(s) with each person of another area by the end of the workshop. Using hypergraph amalgamation-detachment, we prove a more general theorem. In particular we show that above meetings can be scheduled if: , and .

A hypergraph is a pair where is a finite set called the vertex set, is the edge multiset, where every edge is itself a multi-subset of . This means that not only can an edge occur multiple times in , but also each vertex can have multiple occurrences within an edge. The total number of occurrences of a vertex among all edges of is called the degree, of in . For , is said to be -uniform if for each . For , an -factor in a hypergraph is a spanning -regular sub-hypergraph, and an -factorization is a partition of the edge set of into where is an -factor for . We abbreviate -factorization to -factorization.

The hypergraph with (by we mean the collection of all -subsets of ) is called a complete -uniform hypergraph. In connection with Kirkman’s schoolgirl problem [14], Sylvester conjectured that is 1-factorable if and only if . This conjecture was finally settled by Baranyai [8]. Let denote the 3-uniform hypergraph with vertex partition , so that for , and with edge set . One may notice that finding an -factorization for is equivalent to scheduling the meetings between mathematicians with the above restrictions for the case .

If we replace every edge of by copies of , then we denote the new hypergraph by . In this paper, the main result is the following theorem which is obtained by proving a more general result (see Theorem 3.1) using amalgamation-detachment techniques.

Theorem 1.1.

is -factorable if

  1. for ,

  2. for , and

  3. .

In particular, by letting in Theorem 1.1, we solve the Mathematicians Collaboration Problem in the following case.

Corollary 1.2.

is -factorable if

  1. ,

  2. , and

  3. .

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 problem of factorization of complete uniform multipartite hypergraphs, but here we aim to solve this problem under a different notion of “multipartite”. In Baranyai’s definition, an edge can have at most one vertex from each part, but here we allow an edge to have two vertices from each part (see the definition of above). More precise definitions together with preliminaries are given in Section 2, the main result is proved in Section 3, and related open problems are discussed in the last section.

Amalgamation-detachment technique was first introduced by Hilton [10] (who found a new proof for decompositions of complete graphs into Hamiltonian cycles), and was more developed by Hilton and Rodger [11]. Hilton’s method was later genealized to arbitrary graphs [5], and later to hypergraphs [1, 2, 7, 4] leading to various extensions of Baranyai’s theorem (see for example [1, 3]). The results of the present paper, mainly relies on those from [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 of this paper, all hypergraphs (except when we use the term graph) are 3-uniform, so an edge is always of one of the forms , and which we will abbreviate to , and , respectively. In a hypergraph , denotes the multiplicity; for example is the multiplicity of an edge of the form . Similarly, for a graph , is the multiplicity of the edge . A k-edge-coloring of a hypergraph is a mapping , and the sub-hypergraph of induced by color is denoted by . Whenever it is not ambiguous, we drop the subscripts, and also we abbreviate to , to , etc..

Factorizations of the complete graph, , is studied in a very general form in [12, 13], however for the purpose of this paper, a -fold version is needed:

Theorem 2.1.

(Bahmanian, Rodger [6, Theorem 2.3]) is -factorable if and only if is even for and .

Let denote the 3-uniform hypergraph with vertices in which , and for distinct vertices . A (3-uniform) hypergraph is -partite, if there exists a partition of such that for every , for some with . For example, both and are -partite. We need another simple but crucial lemma:

Lemma 2.2.

If is even for , and , then is -factorable.

Proof.

Let with vertex set . By Theorem 2.1, is -factorable. Using this factorization, we obtain a -edge-coloring for such that for every and every color . Now we form a -edge-colored hypergraph with vertex set such that for every pair of distinct vertices , and each color . It is easy to see that and for every and every color . Thus we obtain a -factorization for . ∎

If the multiplicity of a vertex in an edge is , we say that is incident with distinct hinges, say , and we also say that is incident with . The set of all hinges in incident with is denoted by ; so is in fact the degree of .

Intuitively speaking, an -detachment of a hypergraph is a hypergraph obtained by splitting a vertex into one or more vertices and sharing the incident hinges and edges among the subvertices. That is, in an -detachment of in which we split into and , an edge of the form in will be of the form in for some , . Note that a hypergraph and its detachments have the same hinges. Whenever it is not ambiguous, we use , , etc. for degree, multiplicity and other hypergraph parameters in .

Let us fix a vertex of a -edge-colored hypergraph . For , let be the set of hinges each of which is incident with both and an edge of color (so ). For any edge , let be the collection of hinges incident with both and . Clearly, if is of color , then .

A family of sets is laminar if, for every pair of sets belonging to , either , or , or . We shall present two lemmas, both of which follow immediately from definitions.

Lemma 2.3.

Let . Then is a laminar family of subsets of .

For each , and each , let be the set of hinges each of which is incident with both and an edge of the form in (so ).

Lemma 2.4.

Let . Then is a laminar family of disjoint subsets of .

If are real numbers, then and denote the integers such that , and means . We need the following powerful lemma:

Lemma 2.5.

(Nash-Williams [15, Lemma 2]) If are two laminar families of subsets of a finite set , and , then there exist a subset of such that

3. Proofs

Notice that is a -regular hypergraph with vertices and edges. To prove Theorem 1.1, we prove the following seemingly stronger result.

Theorem 3.1.

Let and for , and . Then for all there exists a -edge-colored -vertex -partite hypergraph and a function such that the following conditions are satisfied:

  1. for each part of ;

  2. for each pair of vertices from different parts of ;

  3. for each pair of distinct vertices from the same part, and from a different part of ;

  4. for each color and each .

Remark 3.2.

It is implicitly understood that every other type of edge in is of multiplicity 0.

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 in Theorem 3.1. Then there exists an -partite hypergraph of order and a function such that by (C1) for each part of . This implies that for each and that each part of has vertices. By (C2), for each pair of vertices from different parts of , and by (C3), for each pair of vertices from the same part and from a different part of . This implies that . Finally, by (C4), admits a -edge-coloring such that for each color . This completes the proof. ∎

The idea of the proof of Theorem 3.1 is that each vertex will be split into vertices and 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 .

First we prove the basis of induction, case . Let be and let for all . Since has vertices, it is -partite (each vertex being a partite set). Obviously, for each part of . Also, for each pair of vertices from distinct parts of , so (C2) is satisfied. Since there is only one vertex in each part, (C3) is trivially satisfied.

Since for , and , by Lemma 2.2, is -factorable. Thus, we can find a -edge-coloring for such that for , and therefore (C4) is satisfied.

Suppose now that for some , there exists a -edge-colored -partite hypergraph of order and a function satisfying properties (C1)–(C4) from the statement of the theorem. We shall now construct an -partite hypergraph of order and a function satisfying (C1)–(C4).

Since , is -partite and (C1) holds for , there exists a vertex of with . The graph will be constructed as an -detachment of with the help of laminar families

and

By Lemma 2.5, there exists a subset of such that

(1)

Let with be the hypergraph obtained from by splitting into two vertices and in such a way that hinges which were incident with in become incident in with or according to whether they do not or do belong to , respectively. More precisely,

(2)

So is an -detachment of and the colors of the edges are preserved. Let so that , and for each . It is obvious that is of order , -partite, and for each part of (the new vertex belongs to the same part of as belongs to). Moreover, it is clear that satisfies (C2)–(C4) if . For the rest of the argument, we will repeatedly use the definitions of , (1), and (2).

For we have

so satisfies (C4).

Let so that and (or ) belong to different parts of . We have

Recall that , and for every and , , and thus . This implies that

and so . Now we have

Therefore satisfies (C2).

Let so that belong to different parts of , belong to the same part of , and . We have

Finally, let so that belong to the same part of , and belong to different parts of , and . By an argument very similar to the one above, we have

Therefore satisfies (C3), and the proof is complete. ∎

4. Final Remarks

We define similar to with the difference that in we allow different parts to have different sizes. It seems reasonable to conjecture that

Conjecture 4.1.

is -factorable if and only if

  1. for ,

  2. for , and

  3. .

We prove the necessity as follows. Since is factorable, it must be regular. Let and be two vertices from two different parts, say and parts respectively. Then we have the following sequence of equivalences:

This proves (i). The existence of an -factor implies that for . Since each -factor is an -regular spanning sub-hypergraph and is -regular, we must have .

In Theorem 1.1, we made partial progress toward settling Conjecture 4.1, however at this point, 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 Šajna, and the anonymous referee for their constructive comments.

References

  • [1] M.A. Bahmanian, Detachments of hypergraphs I: The Berge-Johnson problem. Combin. Probab. Comput. 21 (2012), no. 4, 483–495.
  • [2] M.A. Bahmanian, Detachments of amalgamated 3-uniform hypergraphs factorization consequences, J. Combin. Designs 20 (2012), 527–549.
  • [3] M.A. Bahmanian, Connected Baranyai’s theorem. Combinatorica 34 (2014), no. 2, 129–138.
  • [4] M.A. Bahmanian, Mike Newman, Embedding Factorizations for 3-Uniform Hypergraphs II: -factorizations into -factorizations, Electron. J. Combin. 23 (2016), no. 2, Paper 2.42, 14 pp.
  • [5] M.A. Bahmanian, C.A. Rodger, Multiply balanced edge colorings of multigraphs. J. Graph Theory 70 (2012), no. 3, 297–317.
  • [6] M.A. Bahmanian, C.A. Rodger, What are graph amalgamations? Recent Results in Designs and Graphs: a Tribute to Lucia Gionfriddo, Quaderni di matematica 28, University of Naples, 2013, 63–82.
  • [7] M.A. Bahmanian, C.A. Rodger, Embedding factorizations for 3-uniform hypergraphs. J. Graph Theory 73 (2013), no. 2, 216–224.
  • [8] Zs. Baranyai, On the factorization of the complete uniform hypergraph, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I, pp. 91-108. Colloq. Math. Soc. Janos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.
  • [9] Zs. Baranyai, The edge-coloring of complete hypergraphs I, J. Combin. Theory B 26 (1979), no. 3, 276–294.
  • [10] A.J.W. Hilton, Hamilton decompositions of complete graphs, J. Combin. Theory B 36 (1984), 125–134.
  • [11] A.J.W. Hilton, C.A. Rodger, Hamilton decompositions of complete regular -partite graphs, Discrete Math. 58 (1986), 63–78.
  • [12] M. Johnson, Amalgamations of factorizations of complete graphs, J. Combin. Theory Ser. B 97 (2007), no. 4, 597–611
  • [13] W.R. Johnstone, Decompositions of complete graphs, Bull. London Math. Soc. 32 (2000), no. 2, 141–145
  • [14] T.P Kirkman, On a problem in combinations, Camb. Dublin Math. J. 2 (1847), 191–204.
  • [15] C.St.J.A. Nash-Williams, Amalgamations of almost regular edge-colourings of simple graphs, J. Combin. Theory B 43 (1987) 322–342.