Hypergraph matchings, Latin squares, and rainbow matchings

In Chapter 5 we study edge-coloured multigraphs, where each of the colour classes induces a disjoint union of cliques. We determine a condition on the sizes of these colour classes such that a rainbow matching that uses all colours is guaranteed. The motivation for studying this problem arises from different areas of mathematics. In this subsection we summarise previous combinatorial results related to our work. To follow up seamlessly on the previous subsection, let us start with the problem of determining conditions for sparse hypergraphs to contain perfect matchings.

Hypergraph matchings

Amatchingin a hypergraphH = (V, E) is defined as a subsetM ofEsuch that all hyperedges inM are pairwise disjoint. We call a hypergraph matching perfect if it covers all vertices of the hypergraph.

In graphs, maximum matchings can be found efficiently, for instance using Edmonds’

algorithm [74]. In the setting of hypergraphs the problem seems to be more difficult. The decision problem whether, given an integerk, a 3-uniform hypergraphH contains a matching of size at leastkis known to be NP-complete [102]. Hence it is natural to search for conditions that imply the existence of a perfect matching in a hypergraph. We would like to mention the following local resilience result of hypergraphs with respect to perfect matchings. For further results on minimum degree conditions for perfect matchings in hypergraphs we refer to the survey [161] by Zhao.

It was proved by Khan [103] and independently by K¨uhn, Osthus, and Treglown [123]

that a minimum degree strictly greater than ⁿ⁻₂¹

− ^2n/3₂

guarantees a perfect matching in a 3-uniform hypergraph onnvertices ifnis divisible by 3 and sufficiently large. They showed that this bound is actually tight. However, it is believed that if one imposes an additional

1.2. Historical background 13

restriction that forces the hypergraph to look somehow ‘regular’, a much smaller bound on the minimum degree suffices to guarantee a perfect matching.

Indeed, one of the most intriguing conjectures on 3-partite 3-uniform hypergraphs by Ryser [146] suggests that, ifnis odd, a minimum degree of n suffices to guarantee a perfect matching in a balanced 3-partite 3-uniform hypergraph on 3nvertices if every pair of vertices from different partition classes lies in exactly one hyperedge. If n is even, Brualdi [46] and independently Stein [148] conjectured that such a hypergraph contains a matching of sizen−1.

These conjectures were originally formulated in terms of Latin squares. Before returning to the just mentioned conjecture, we will first introduce Latin squares.

Latin squares

A Latin square of order n is an n×n matrix filled with n different symbols such that each symbol appears exactly once in every row and exactly once in every column. There is a one-to-one correspondence between Latin squares and balanced 3-partite hypergraphs with each pair of vertices from different partition classes being contained in exactly one hyperedge.

Simply let the rows, columns, and symbols of a Latin square define the partition classes of a 3-partite hypergraph and let the set of hypergraphs consist of those triples {x, y, z}, where the symbol z appears in row x and column y. In the same way one can construct a unique Latin square for every hypergraph with the stated property.

The study of Latin squares has a long history and was first systematically developed by Euler (see e.g. [61]). Latin squares have various applications in different branches of mathematics (see [61] for an extensive survey); for instance in algebra, where Latin squares are the multiplication tables of quasigroups, and in a branch of statistics called ‘design of experiments’, where Latin squares are a special case of row-column designs for two blocking factors. Possibly many people come across Latin squares in recreational mathematics as completed Sudoku puzzles are Latin squares, typically of order 9.

A matching in a hypergraph H, which is associated with a Latin square L, corresponds to a so-called partial transversal in L. A partial transversal in a Latin square is a set of entries with distinct symbols such that from each row and each column at most one entry is contained in this set. We call a partial transversal of size n in a Latin square of order n simplytransversal.

A motivation to study transversals areorthogonal Latin squares, where two Latin squares (A_i,j)_i,j∈[n] and (B_i,j)_i,j∈[n] are called orthogonal if all pairs {(A_i,j, B_i,j)}i,j∈[n] are distinct.

Orthogonal Latin squares, also known under the name Graeco-Latin squares, are used for instance in experimental design and tournament scheduling (see e.g. [61]). It can be seen fairly quickly that a Latin square has an orthogonal mate if and only if it has a decomposition into disjoint transversals.

By considering Latin squares of order 2 one can easily verify that not every Latin square has a transversal. For every evenn∈N, the cyclic Latin square of ordern (i.e. the addition table of the group of integers modulon) does not have a transversal (see e.g. [156]). For every odd integern ≥11, however, it is still an open question whether a Latin square of order n without a transversal exists. This brings us back to the famous conjecture by Ryser [146].

Conjecture 1.11 (Ryser [146]). Every Latin square of odd order has a transversal.

Observe that this is an equivalent form of the conjecture that we mentioned above in terms of hypergraph matchings. Conjecture 1.11 is known to be true forn≤9 (see [128]). For all

n∈N, Brualdi [46] conjectured that every Latin square of ordernhas a partial transversal of size n−1. Independently, Stein [148] made the stronger conjecture that every n×nmatrix that is filled with nsymbols each appearing exactlyn times contains a partial transversal of size n−1. Because of the similarity of these conjectures, the following one is often referred to as the Brualdi-Stein conjecture.

Conjecture 1.12 (Brualdi-Stein [46, 148]). For every n≥1any Latin square of ordern has a partial transversal of size n−1.

Recall that we have already mentioned above an equivalent form of Conjecture 1.12 in the setting of hypergraphs. There have been several approaches to Conjecture 1.12. For instance Hatami and Shor [90] proved that every Latin square contains a partial transversal of length n− O(log²n). This improves on earlier results by Brouwer, de Vries, Wieringa [45]

and Woolbright [158], who showed independently that every Latin square contains a partial transversal of size n−√

n, and by Drake [64] and Koksma [107], who determined the lower bounds 3n/4 and 2n/3, respectively.

There is yet another way to rephrase Conjectures 1.11 and 1.12, namely to the study of rainbow matchings in bipartite edge-coloured graphs. This also allows a more general setting for strengthenings of the above conjectures. For more details on Latin squares we refer to the survey [155] by Wanless.

Rainbow matchings

As already indicated, a natural way to transfer Conjectures 1.11 and 1.12 to graphs is the following. Let L = (L_i,j)_i,j∈[n] be a Latin square of order n. We define G_L := (A∪B, E) as the complete bipartite edge-coloured graph with partition classes A = {a1, . . . , an} and B={b₁, . . . , b_n}, where {a_i, b_j} is coloured with colour L_i,j. That is,A and B represent the columns and rows of L, respectively. Moreover, a transversal of L corresponds to a perfect matching in GL that uses each edge colour exactly once. Such a matching is called rainbow matching of size n. Using this notion, Conjecture 1.12 is equivalent to the following: For every n ≥ 1 any complete bipartite edge-coloured graph, the colour classes of which are perfect matchings, contains a rainbow matching of size n−1. It is believed that this is true in the more general setting of bipartite edge-coloured multigraphs. Indeed, Aharoni and Berger [2] conjectured the following generalisation of Conjecture 1.12.

Conjecture 1.13 (Aharoni, Berger [2]). Let G be a bipartite multigraph, the edges of which are coloured withn colours and such that each colour class induces a matching of sizen+ 1.

Then there is a rainbow matching of size n.

While Conjecture 1.13 remains open, asymptotic versions are known to be true. For instance, Barat, Gy´arf´as, and S´ark¨ozy [19] extended Woolbright’s arguments to multigraphs proving that every bipartite edge-coloured multigraph, each of whosencolour classes has size at least n, contains a rainbow matching withn−√

nedges.

Another possibility to approach Conjecture 1.13 is to let the colour classes be bigger than nwhile keeping the requirement of the rainbow matching to be of size n. Aharoni, Charbit, and Howard [3] proved that sizes of b7n/4c, and Kotlar and Ziv [113] proved that sizes of b5n/3c suffice to guarantee a rainbow matching of size n. In Chapter 5 we further improve this bound.

1.2. Historical background 15

It is worthwhile mentioning that when the matchings induced by the colour classes are all edge-disjoint, a theorem by H¨aggkvist and Johansson [88] on so-called Latin rectangles implies that there is a rainbow matching of sizenin case when all colour classes induce edge-disjoint perfect matchings of size n+o(n). Pokrovskiy [138] provided a proof for the more general case where the matchings are not necessarily perfect and thus proved an approximate version of Conjecture 1.13 in the case when the matchings are edge-disjoint.