Partial Transversals Research Articles

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 $n.$

Latin squares with maximal partial transversals of many lengths

A partial transversal T of a Latin square L is a set of entries of L in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any maximal partial transversal of a Latin square of order n has size at least ⌈n2⌉ and at most n. We say that a Latin square is omniversal if it possesses a maximal partial transversal of all plausible sizes and is near-omniversal if it possesses a maximal partial transversal of all plausible sizes except one.Evans (2019) showed that omniversal Latin squares of order n exist for any odd n≠3. By extending this result, we show that an omniversal Latin square of order n exists if and only if n∉{3,4} and n≢2(mod4). Furthermore, we show that near-omniversal Latin squares exist for all orders n≡2(mod4).Finally, we show that no non-trivial finite group has an omniversal Cayley table, and only 15 finite groups have a near-omniversal Cayley table. In fact, as n grows, Cayley tables of groups of order n miss a constant fraction of the plausible sizes of maximal partial transversals. In the course of proving this, we partially solve the following interesting problem in combinatorial group theory. Suppose that we have two finite subsets R,C⊆G of a group G such that |{rc:r∈R,c∈C}|=m. How large do |R| and |C| need to be (in terms of m) to be certain that R⊆xH and C⊆Hy for some subgroup H of order m in G, and x,y∈G?

New bounds for the Moser‐Tardos distribution

The Lovász local lemma (LLL) is a probabilistic tool to generate combinatorial structures with good “local” properties. The “LLL‐distribution” further shows that these structures have good global properties in expectation. The seminal algorithm of Moser and Tardos turned the simplest, variable‐based form of the LLL into an efficient algorithm; this has since been extended to other probability spaces including random permutations. One can similarly define an “MT‐distribution” for these algorithms, that is, the distribution of the configuration they produce. We show new bounds on the MT‐distribution in the variable and permutation settings which are significantly stronger than those known to hold for the LLL‐distribution. As some example illustrations, we show a nearly tight bound on the minimum implicate size of a CNF Boolean formula, and we obtain improved bounds on weighted Latin transversals and partial Latin transversals.

The Chromatic Number of Finite Group Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture—commonly attributed to Brualdi—that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.

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 }.

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Moser and Tardos have developed a powerful algorithmic approach (henceforth MT) to the Lovász Local Lemma (LLL); the basic operation done in MT and its variants is a search for “bad” events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables that arise during intermediate stages of MT. We show that these configurations have a more or less “random” form, building further on the MT-distribution concept of Haeupler et al. in understanding the (intermediate and) output distribution of MT. This has a variety of algorithmic applications; the most important is that bad events can be found relatively quickly, improving on MT across the complexity spectrum. It makes some polynomial-time algorithms sublinear (e.g., for Latin transversals, which are of basic combinatorial interest), gives lower-degree polynomial runtimes in some settings, transforms certain superpolynomial-time algorithms into polynomial-time algorithms, and leads to Las Vegas algorithms for some coloring problems for which only Monte Carlo algorithms were known. We show that, in certain conditions when the LLL condition is violated, a variant of the MT algorithm can still produce a distribution that avoids most of the bad events. We show in some cases that this MT variant can run faster than the original MT algorithm itself and develop the first-known criterion for the case of the asymmetric LLL. This can be used to find partial Latin transversals—improving on earlier bounds of Stein (1975)—among other applications. We furthermore give applications in enumeration, showing that most applications (for which we aim for all or most of the bad events to be avoided) have large solution sets. We do this by showing that the MT distribution has large Rényi entropy.

ON THE CHROMATIC INDEX OF LATIN SQUARES

A proper coloring of a Latin square of order n is an assignment of colors to its elements triples such that each row, column and symbol is assigned n distinct colors. Equivalently, a proper coloring of a Latin square is a partition into partial transversals. The chromatic index of a Latin square is the least number of colors needed for a proper coloring. We study the chromatic index of the cyclic Latin square which arises from the addition table for the integers modulo n . We obtain the best possible bounds except for the case when n/ 2 is odd and divisible by 3. We make some conjectures about the chromatic index, suggesting a generalization of Ryser’s conjecture (that every Latin square of odd order contains a transversal).

On the chromatic number of Latin square graphs

The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some other classes of Latin squares. With a computer, we find the chromatic number for all main classes of Latin squares of order at most eight.

Extensions and presentations of transversal matroids

A transversal matroid M can be represented by a collection of sets, called a presentation of M, whose partial transversals are the independent sets of M. Minimal presentations are those for which removing any element from any set gives a presentation of a different matroid. We study the connections between (single-element) transversal extensions of M and extensions of presentations of M. We show that a presentation of M is minimal if and only if different extensions of it give different extensions of M; also, all transversal extensions of M can be obtained by extending the minimal presentations of M. We also begin to explore the partial order that the weak order gives on the transversal extensions of M.

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).

Two Results on Infinite Transversal Theory

By the assistance of an independence space on a set S with finite rank, this paper presents a way to determine a family of subsets of S to possess an independent transversal. Afterwards, it provides another result on infinite transversal relative to partial transversals of a family of subsets of S. The two results are in the field of transversal theory. Keywords: transversal; independence space; finite rank; finite partial transversal. 2010 Mathematics Subject Classification: 05B35; 52A35

Existence of partial transversals

We use the polynomial method to study the existence of partial transversals in the Cayley addition table of Abelian groups.

Dual-bounded generating problems: weighted transversals of a hypergraph

We consider a generalization of the notion of transversal to a finite hypergraph, the so-called weighted transversals. Given a non-negative weight vector assigned to each hyperedge of an input hypergraph A and a non-negative threshold vector, we define a weighted transversal as a minimal vertex set which intersects all the hyperedges of A except for a sub-family of total weight not exceeding the given threshold vector. Weighted transversals generalize partial and multiple transversals introduced in Boros et al. (SIAM J. Comput. 30 (6) (2001)) and also include minimal binary solutions to non-negative systems of linear inequalities and minimal weighted infrequent sets in databases. We show that the hypergraph of all weighted transversals is dual-bounded, i.e., the size of its transversal hypergraph is polynomial in the number of weighted transversals and the size of the input hypergraph. Our bounds are based on new inequalities of extremal set theory and threshold Boolean logic, which may be of independent interest. For instance, we show that for any row-weighted m× n binary matrix and any threshold weight t, the number of maximal sets of columns whose row support has weight above t is at most m times the number of minimal sets of columns with row support of total weight below t. We also prove that the problem of generating all weighted transversals for a given hypergraph is polynomial-time reducible to the generation of all ordinary transversals for another hypergraph, i.e., to the well-known hypergraph dualization problem. As a corollary, we obtain an incremental quasi-polynomial-time algorithm for generating all weighted transversals for a given hypergraph. This result includes as special cases the generation of all the minimal Boolean solutions to a given system of non-negative linear inequalities and the generation of all minimal weighted infrequent sets of columns for a given binary matrix.

Dual-Bounded Generating Problems: Partial and Multiple Transversals of a Hypergraph

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistoryPublished online: 27 July 2006Keywordstransversal, minimal infrequent set, independent set, hypergraph, dualization, monotone Boolean function, incremental polynomial time, threshold function, data mining, maximal frequent setAMS Subject Headings68R05, 68Q25, 68Q32Publication DataISSN (print): 0097-5397ISSN (online): 1095-7111Publisher: Society for Industrial and Applied MathematicsCODEN: smjcat

Symmetrized and continuous generalization of transversals

The theorem of Edmonds and Fulkerson states that the partial transversals of a finite family of sets form a matroid. The aim of this paper is to present a symmetrized and continuous generalization of this theorem.

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

Maximal sets of Latin squares and partial transversals

Maximal sets of Latin squares and partial transversals

On P-quasigroups and decompositions of complete undirected graphs

In [2], A. Kotzig has introduced the concepts of P-groupoid and P-quasigroup and has shown how these concepts are related to the decomposition of a complete undirected graph into disjoint closed paths. To each closed path of the graph associated with a given P-quasigroup Q there corresponds a cyclic partial transversal in the Latin square L which is defined by the multiplication table of Q. In this paper, it is shown that cyclic transversals are connected with Hamiltonian decompositions of complete undirected graphs having an even number of vertices and a connection between the order of a particular type of P-quasigroup and the length of its cyclic partial transversals is established. An indirect connection with the work of Yap [4] is established via the concept of isotopy.

Common partial transversals and integral matrices

Certain packing and covering problems associated with the common partial transversals of two families $\mathfrak {A}$ and $\mathfrak {B}$ of subsets of a set $E$ are investigated. Under suitable finitary restrictions, necessary and sufficient conditions are obtained for there to exist pairwise disjoint sets ${F_1}, \ldots ,{F_t}$ where each ${F_i}$ is a partial transversal of $\mathfrak {A}$ with defect at most $p$ and a partial transversal of $\mathfrak {B}$ with defect at most $q$. We also prove that (i) $E = \cup _{i = 1}^t{T_i}$ where each ${T_i}$ is a common partial transversal of $\mathfrak {A}$ and $\mathfrak {B}$ if and only if (ii) $E = \cup _{i = 1}^t{T_i}’$ where each ${T_i}’$ is a partial transversal of $\mathfrak {A}$ and (iii) $E = \cup _{i = 1}^t{T_i}''$ where each ${T_i}''$ is a partial transversal of $\mathfrak {B}$. We then derive necessary and sufficient conditions for the validity of (i). The proofs are accomplished by establishing a connection with these common partial transversal problems and representations of integral matrices (not necessarily finite or countably infinite) as sums of subpermutation matrices and then using known results about the existence of a single common partial transversal of two families. Accordingly various representation theorems for integral matrices are derived.