Transversals Of Latin Squares Research Articles

Rainbow independent sets in certain classes of graphs

AbstractRainbow matchings in graphs and in hypergraphs have been studied extensively, one motivation coming from questions on matchings in 3‐partite hypergraphs, including questions on transversals in Latin squares. Matchings in graphs are independent sets in line graphs, so a natural problem is to extend the study to rainbow independent sets in general graphs. We study problems of the following form: given a class of graphs, how many independent sets of size in a graph belonging to are needed to guarantee the existence of a rainbow set of size ? A particularly interesting case is the class of graphs having a given upper bound on their maximum degree.

Pursuit-evasion games on Latin square graphs

We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number. The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given. If $n>(k+1)^2,$ then the cop number is shown to be $k+2.$ Lower and upper bounds are provided for the metric dimension and localization number of latin square graphs. The metric dimension of back-circulant latin squares shows that the lower bound is close to tight. Recent results on covers and partial transversals of latin squares provide the upper bound of $n+O\left(\frac{\log{n}}{\log{\log{n}}}\right)$ on the localization number of a latin square graph of order $n.$

Almost color-balanced perfect matchings in color-balanced complete graphs

For a graph G and a not necessarily proper k-edge coloring c:E(G)→{1,…,k}, let mi(G) be the number of edges of G of color i, and call G color-balanced if mi(G)=mj(G) for every two colors i and j. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly n-edge colored complete bipartite graph Kn,n has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M. For a perfect matching M of K2kn, a natural measure for the total deviation of M from being color-balanced is f(M)=∑i=1k|mi(M)−n|. While not every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M, that is, a perfect matching with f(M)=0, we prove the existence of a perfect matching M with f(M)=O(kknln⁡(k)) for general k and f(M)≤2 for k=3; the case k=2 has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.

Small Latin arrays have a near transversal

AbstractA Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an n × n array is a selection of cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order has a diagonal of weight at least . A corollary is the existence of near transversals in Latin squares of these orders. More generally, for all we compute a lower bound on the order of any Latin array that does not have a diagonal of weight at least .

Long directed rainbow cycles and rainbow spanning trees

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. The problem of finding rainbow subgraphs goes back to the work of Euler on transversals in Latin squares and was extensively studied since then. In this paper we consider two related questions concerning rainbow subgraphs of complete, edge-coloured graphs and digraphs. In the first part, we show that every properly edge-coloured complete directed graph contains a directed rainbow cycle of length n−O(n4∕5). This is motivated by an old problem of Hahn and improves a result of Gyarfas and Sarkozy. In the second part, we show that any tree T on n vertices with maximum degree ΔT≤βn∕logn has a rainbow embedding into a properly edge-coloured Kn provided that every colour appears at most αn times and α,β are sufficiently small constants.

Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv2\bmod4$.

Differences of Bijections

When is a function from an abelian group to itself expressible as a difference of two bijections? Answering this question for finite cyclic groups solves a problem about juggling. A theorem of Marshall Hall settles the question for finite abelian groups, and a forgotten theorem of László Fuchs settles the question for infinite abelian groups. After explicating these theorems, we extend the problem by examining expressibility as a difference of injections or surjections. We also extend the question beyond the realm of group theory, where the expressibility of a function translates to questions about partial transversals in Latin squares.

Covers and partial transversals of Latin squares

We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order n is n+a if and only if the maximum size of a partial transversal is either n-2a or n-2a+1. (2) A minimal cover in a Latin square of order n has size at most mu _n=3(n+1/2-sqrt{n+1/4}). (3) There are infinitely many orders n for which there exists a Latin square having a minimal cover of every size from n to mu _n. (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to mu _n. (5) If 1leqslant kleqslant n/2 and ngeqslant 5 then there is a Latin square of order n with a maximal partial transversal of size n-k. (6) For any varepsilon >0, asymptotically almost all Latin squares have no maximal partial transversal of size less than n-n^{2/3+varepsilon }.

An approximate version of a conjecture of Aharoni and Berger

Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by n colours with at least n+1 edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding problem of Brualdi and Stein about transversals in Latin squares. Here an approximate version of the Aharoni–Berger Conjecture is proved—it is shown that if there are at least n+o(n) edges of each colour in a proper n-edge-colouring of a bipartite multigraph then there is a rainbow matching using every colour.

Rainbow Matchings and Rainbow Connectedness

Aharoni and Berger conjectured that every collection of $n$ matchings of size $n+1$ in a bipartite graph contains a rainbow matching of size $n$. This conjecture is related to several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. The conjecture is known to hold when the matchings are much larger than $n+1$. The best bound is currently due to Aharoni, Kotlar, and Ziv who proved the conjecture when the matchings are of size at least $3n/2+1$. When the matchings are all edge-disjoint and perfect, the best result follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least $n+o(n)$. In this paper we show that the conjecture is true when the matchings have size $n+o(n)$ and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least $\phi n+o(n)$ where $\phi\approx 1.618$ is the Golden Ratio.Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

Loops with exponent three in all isotopes

It was shown by van Rees [Subsquares and transversals in latin squares, Ars Combin. 29B (1990) 193–204] that a latin square of order [Formula: see text] has at most [Formula: see text] latin subsquares of order [Formula: see text]. He conjectured that this bound is only achieved if [Formula: see text] is a power of [Formula: see text]. We show that it can only be achieved if [Formula: see text]. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent [Formula: see text]. We call such loops van Rees loops and show that they form an equationally defined variety. We also show that: (1) In a van Rees loop, any subloop of index 3 is normal. (2) There are exactly six nonassociative van Rees loops of order [Formula: see text] with a nontrivial nucleus and at least 1 with all nuclei trivial. (3) Every commutative van Rees loop has the weak inverse property. (4) For each van Rees loop there is an associated family of Steiner quasigroups.

Rainbow matchings and connectedness of coloured graphs

Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n+1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. When the matchings are all edge-disjoint and perfect, an approximate version of this conjecture follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least n+o(n).Here we'll discuss a proof of this conjecture in the case when the matchings have size n+o(n) and are all edge-disjoint (but not necessarily perfect). The proof involves studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

Multidimensional Permanents and an Upper Bound on the Number of Transversals in Latin Squares

AbstractThe permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to . Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares urn:x-wiley:10638539:media:jcd21413:jcd21413-math-0003

Rainbow matchings and cycle-free partial transversals of Latin squares

In this paper we show that properly edge-colored graphs G with |V(G)|≥4δ(G)−3 have rainbow matchings of size δ(G); this gives the best known bound for a recent question of Wang. We also show that properly edge-colored graphs G with |V(G)|≥2δ(G) have rainbow matchings of size at least δ(G)−2δ(G)2/3. This result extends (with a weaker error term) the well-known result that a factorization of the complete bipartite graph Kn,n has a rainbow matching of size n−o(n), or equivalently that every Latin square of order n has a partial transversal of size n−o(n) (an asymptotic version of the Ryser–Brualdi conjecture). In this direction we also show that every Latin square of order n has a cycle-free partial transversal of size n−o(n).

Transversals of Latin squares and covering radius of sets of permutations

We consider the symmetric group Sn as a metric space with the Hamming metric. The covering radius cr(S) of a set of permutations S⊂Sn is the smallest r such that Sn is covered by the balls of radius r centred at the elements of S. For given n and s, let f(n,s) denote the cardinality of the smallest set S of permutations with cr(S)⩽n−s.The value of f(n,2) is the subject of a conjecture by Kézdy and Snevily that implies two famous conjectures by Ryser and Brualdi on transversals in Latin squares. We show that f(n,2)⩽n+O(logn) for all n and that f(n,2)⩽n+2 whenever n=3m for m>1. We also construct, for each odd m⩾3, a Latin square of order 3m with two rows that each contain 2m−2 transversal-free entries. This gives an infinite family of Latin squares with odd order n and at most n/3+O(1) disjoint transversals. The previous strongest upper bound for such a family was n/2+O(1).Finally, we show that f(5,3)=15 and record a proof by Blackburn that cr(AGL(1,q))=q−3 when q is odd.

Graceful Labelling: State of the Art, Applications and Future Directions

This paper takes a close look at graceful labelling and its applications. We pay special attention to the famous Graceful Tree Conjecture, which has attracted a lot of interest and engaged many researchers over the last 40+ years, and yet to this day remains unsolved. We describe applications of graceful and graceful-like labellings of trees to several well known combinatorial problems and we expose yet another one, namely the connection between α-labelling of paths and near transversals in Latin squares. Finally, we show how spectral graph theory can be used to further the progress on the Graceful Tree Conjecture.

The number of transversals in a Latin square

A Latin Square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin Square of order n such that no two entries contain the same symbol. Define T(n) to be the maximum number of transversals over all Latin squares of order n. We show that $$b^n \leq T(n) \leq c^n\sqrt{n}\,n!$$ for n ? 5, where b ? 1.719 and c ? 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n × n toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin Squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.

Covering radius for sets of permutations

We study the covering radius of sets of permutations with respect to the Hamming distance. Let f ( n , s ) be the smallest number m for which there is a set of m permutations in S n with covering radius r ⩽ n - s . We study f ( n , s ) in the general case and also in the case when the set of permutations forms a group. We find f ( n , 1 ) exactly and bounds on f ( n , s ) for s > 1 . For s = 2 our bounds are linear in n. This case relates to classical conjectures by Ryser and Brualdi on transversals of Latin squares and to more recent work by Kézdy and Snevily. We discuss a flaw in Derienko's published proof of Brualdi's conjecture. We also show that every Latin square contains a set of entries which meets each row and column exactly once while using no symbol more than twice. In the case where the permutations form a group, we give necessary and sufficient conditions for the covering radius to be exactly n. If the group is t-transitive, then its covering radius is at most n - t , and we give a partial determination of groups meeting this bound. We give some results on the covering radius of specific groups. For the group PGL ( 2 , q ) , the question can be phrased in geometric terms, concerning configurations in the Minkowski plane over GF ( q ) meeting every generator once and every conic in at most s points, where s is as small as possible. We give an exact answer except in the case where q is congruent to 1 mod 6. The paper concludes with some remarks about the relation between packing and covering radii for permutations.

Hamilton circuits with many colours in properly edge-coloured complete graphs.

We prove that a properly edge-coloured complete graph K n has a Hamilton circuit with edges of at least n−√2n distinct colours. This is proved with a method inspired by work on long partial transversals in latin squares. Another such method is employed in proving a similar result where the distinct colours all occur on edges not belonging to a given spanning set of edges of K n

Transversals of Latin squares and their generalizations

Transversals of Latin squares and their generalizations