Systems Of Distinct Representatives Research Articles

INDEXING METHODS FOR FORMING COMBINATORIAL CONFIGURATIONS OF THE CLASS OF SYSTEMS OF DISTINCT REPRESENTATIVES

Abstract. Objectives In this paper, methods for solving of a class of combinatorial tasks, known as systems of distinct representatives (SDR), are considered. The objective is to develop methods and algorithms for the formation of a combinatorial SDR configuration that includes as rows, columns or row- and column subsets, which are composed of elements of the original family of nxn – sets occupying different positions in the initial sets, as well to determine the possible number of proposed configurations. Method Index ordering methods are used for the arrangement of elements in the formed systems of distinct representatives, the essence of which is to formulate requirements for the process of configuration having specified properties through the regularity of indexing elements within these configurations. Results The general formulation of the issue of constructing an SDR is considered in terms of a problem of formation from the elements of sets and subsets, which include one element from each initial set, with each of these elements being located at different positions in the original sets. The task was reformulated in reference to the requirements for indexing the elements of these subsets. Each element in set systems has a two-index designation, with the first element in the index indicating membership of a specific initial set and the second – to its location. In order to fulfil the requirements formulated in the task, it is necessary for indices of the SDR elements to have values from 1 to n. Conclusion Two methods for solving the problem are proposed: cyclic shifts of rows and columns of the matrix configuration formed by the original sets, and by a given law for indexing the elements of the environment. The number of possible options for the formation of representative systems is determined. The reasons for the propagation of the proposed methods for solving the problem are established only for initial sets of odd dimensions.

From rankings’ collinearity to counting SDR's via chromatic list expression

The problem of counting rankings satisfying the collinearity condition with respect to two rankings over the metric spaces of rank distance is treated in this paper by transforming it into finding the number of systems of distinct representatives (abbreviated as SDR's) with respect to associated set systems. A formula for the chromatic list expression of graphs is then given in terms of the inclusion and exclusion principle, followed by a formula for the number of SDR's of set systems when the graphs are complete.

A Hall-type theorem for triplet set systems based on medians in trees

Given a collection C of subsets of a finite set X , let ⋃ C = ∪ S ∈ C S . Philip Hall’s celebrated theorem [P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26–30] concerning ‘systems of distinct representatives’ tells us that for any collection C of subsets of X there exists an injective (i.e. one-to-one) function f : C → X with f ( S ) ∈ S for all S ∈ C if and and only if C satisfies the property that for all non-empty subsets C ′ of C , we have | ⋃ C ′ | ≥ | C | . Here, we show that if the condition | ⋃ C ′ | ≥ | C ′ | is replaced by the stronger condition | ⋃ C ′ | ≥ | C ′ | + 2 , then we obtain a characterization of this condition for a collection of 3-element subsets of X in terms of the existence of an injective function from C to the vertices of a tree whose vertex set includes X and which satisfies a certain median condition. We then describe an extension of this result to collections of arbitrary-cardinality subsets of X .

Extending Hall's Theorem into List Colorings: A Partial History

In 1988, A. J. W. Hilton and P. D. Johnson Jr. found a natural generalization of the condition in Philip Hall's celebrated theorem on systems of distinct representatives. This generalization was formed in the relatively new theory of list colorings of graphs. Here we give an account of a strand of development arising from this generalization, concentrating on extensions of Hall's theorem. New results are presented concerning list colorings of independence systems and colorings of graphs with nonnegative measurable functions on positive measure spaces.

Theorems on sets not belonging to algebras

Let A 1 , … , A n , A n + 1 \mathcal {A}_1,\dots , \mathcal {A}_n, \mathcal {A}_{n+1} be a finite sequence of algebras of sets given on a set X X , ⋃ k = 1 n A k ≠ P ( X ) \bigcup _{k=1}^n \mathcal {A}_k \ne \mathfrak {P}(X) , with more than 4 3 n \frac {4}{3}n pairwise disjoint sets not belonging to A n + 1 \mathcal {A}_{n+1} . It has been shown in the author’s previous articles that in this case ⋃ k = 1 n + 1 A k ≠ P ( X ) \bigcup _{k=1}^{n+1} \mathcal {A}_k \ne \mathfrak {P}(X) . Let us consider, instead of A n + 1 \mathcal {A}_{n+1} , a finite sequence of algebras A n + 1 , … , A n + l \mathcal {A}_{n+1}, \dots , \mathcal {A}_{n+l} . It turns out that if for each natural i ≤ l i \le l there exist no less than 4 3 ( n + l ) − l 24 \frac {4}{3}(n+l)- \frac {l}{24} pairwise disjoint sets not belonging to A n + i \mathcal {A}_{n+i} , then ⋃ k = 1 n + l A k ≠ P ( X ) \bigcup _{k=1}^{n+l} \mathcal {A}_k \ne \mathfrak {P}(X) . Besides this result, the article contains: an essentially important theorem on a countable sequence of almost σ \sigma -algebras (the concept of almost σ \sigma -algebra was introduced by the author in 1999), a theorem on a family of algebras of arbitrary cardinality (the proof of this theorem is based on the beautiful idea of Halmos and Vaughan from their proof of the theorem on systems of distinct representatives), a new upper estimate of the function v ( n ) \mathfrak {v}(n) that was introduced by the author in 2002, and other new results.

Transversals, systems of distinct representatives, mechanism design, and matching

Transversals, systems of distinct representatives, mechanism design, and matching

Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs

Philip Hall's famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or list-multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural generalization to a condition we call Hall's condition on a simple graph G, a vertex list assignment to G, and an assignment of nonnegative integers to the vertices of G. Hall's condition turns out to be necessary for the existence of a proper multicoloring of G under these assignments. The Hall-Halmos-Vaughan theorem may be stated: when G is a clique, Hall's condition is sufficient for the existence of a proper multicoloring. In this article, we undertake the study of the class HHV of simple graphs G for which Hall's condition is sufficient for the existence of a proper multicoloring. It is shown that HHV is contained in the class ℋ︁0 of graphs in which every block is a clique and each cut-vertex lies in exactly two blocks. On the other hand, besides cliques, the only connected graphs we know to be in HHV are (i) any two cliques joined at a cut-vertex, (ii) paths, and (iii) the two connected graphs of order 5 in ℋ︁0, which are neither cliques, paths, nor two cliques stuck together. In case (ii), we address the constructive aspect, the problem of deciding if there is a proper coloring and, if there is, of finding one. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 199–219, 2000

A Generalization of Distinct Representatives and Its Applications

A generalization of systems of distinct representatives is given. Based on this result, it is shown bow unequally replicated block designs, satisfying a simple condition, can be arranged in an array such that each row of the array contains a specified treatment the same number of times. Such arrays can be viewed as row-colum designs and have been called as Youden Type ( YT) designs. A necessary and sufficient condition for the existence of YT designs is given.

Counting common representatives and symmetric chain decompositions

SynopsisWe obtain lower bounds for the number of common systems of distinct representatives of two families of sets and the number of symmetric chain decompositions of certain ranked partially ordered sets.

Algorithms for the construction of (M,S)-optimal block designs and row-column designs

Algorithms for the generation of (M, S)optimal block and row-column designs are presented. The first algorithm generates the treatment-block incidence matrix of an (M, S)-optimal block design. The second searches for the particular permutation of the rows of an incidence matrix which gives the best possible combination of rows and columns within the row-column design. A row-column design is then obtained by combining the selected treatment-row and treatment-column arrangements using the method of Systems of Distinct Representatives.

Dividing a cake fairly

If each of n people defines a (suitable) measure on a compact convex cake I, then there exists a division of I into n connected parts, and an assignment of these n parts to the n people, in such a way that the piece of cake assigned to each person is at least as large (in his own measure) as that assigned to anyone else. (The proof uses Brouwer's fixed-point theorem and Hall's theorem on systems of distinct representatives.) Slight progress is made towards finding algorithms for this problem.

Systems of distinct representatives for k families of sets

Systems of distinct representatives for k families of sets

On the number of systems of distinct representatives in an independence structure

On the number of systems of distinct representatives in an independence structure

A Remark on a Theorem of M. Hall

Let A = (aij) be an m×n (0, 1)-matrix, m≤n. The permanent of A is defined bywhere the summation is over all one-one functions σ:{1, 2, . . . , m}→{1, 2, . . . , n}. If A is regarded as the incidence matrix of a configuration of m subsets of an n-set, then Per (A) is the number of systems of distinct representatives in the configuration.

On distinct representatives and mapping theorems

We present an elementary proof of a transfinite symmetric form of Philip Hall's theorem on systems of distinct representatives and derive some wellknown mapping theorems from it to illustrate the ease with which this form of the theorem may be used.

Systems of distinct representatives, II

Systems of distinct representatives, II

A Very General Theorem on Systems of Distinct Representatives

A Very General Theorem on Systems of Distinct Representatives

A problem connected with systems of distinct representatives

A problem connected with systems of distinct representatives

Systems of distinct representatives and linear algebra

Systems of distinct representatives and linear algebra

On the Number of Systems of Distinct Representatives of Sets

On the Number of Systems of Distinct Representatives of Sets