Marriage Theorem Research Articles

Couplings and Matchings: Combinatorial notes on Strassen’s theorem

Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall’s marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory, in the sense that these results can easily be derived from each other. Remarkably, this equivalence extends to Strassen’s theorem, a celebrated result on couplings of probability measures. In this paper the equivalence between Strassen’s theorem and Hall’s theorem is investigated from a combinatorial perspective. A novel combinatorial lemma will be introduced that can be used to deduce both Hall’s theorem and a finite version of Strassen’s theorem, providing a simple proof of their equivalence.

On the Hadamard-Fischer inequality, the inclusion-exclusion formula, and bipartite graphs

The classical Hadamard-Fischer-Koteljanskii inequality is an inequality between principal minors of positive definite matrices. In this work, we present an extension of the Hadamard-Fischer-Koteljanskii inequality, that is inspired by the inclusion-exclusion formula for sets. We formulate necessary and sufficient conditions for the inequality to hold. We describe general structures of the collection of index sets involved. In analyzing these structures, a graph-theoretical property that applies to bipartite graphs is found. We establish that if the vertices of a bipartite graph satisfy simple conditions, then the bipartite graph contains a vertex subgraph which is a cycle or a complete subgraph missing a matching. This result is reminiscent of the Hall's marriage theorem for bipartite graphs.

Research on Matching and Vertex Cover Problems in Bipartite Graphs using Simplex Method

This paper considers a bipartite graph where a perfect matching does not necessarily exist. Linear programming is used in this paper as a special case of the linear program is the assignment problem, which is another name for the weighted maximum matching problem. The objective is to show that linear programming, in particular the simplex algorithm, can be used to calculate maximum weight matchings and minimum weighted vertex covers. This study also calculates and shows the equivalence of the maximum matching and minimum vertex cover cardinalities, and uses linear programming duality to present its relevance to König's theorem. Finally, Hall’s marriage theorem is used to explicitly prove the absence of a perfect matching in particular graphs. There exist many algorithms developed over the years that are designed to solve maximum matching problems in polynomial time, but the simplex method is one of the oldest and simplest algorithms to understand in solving these problems.

Graph Models of Harems and Tournaments in Sports Clubs

By looking at the extension of Hall's marriage theorem to harems, where some people are allowed to have more than one partner, Traditionally in harems any man can have multiple wives but no woman can have more than one husband. then consider the different types of matches by looking at 'round robin tournaments' in sports clubs. An unexpected connection between the two worlds emerged when we were able to use our harem results to deduce theorems about the tournament.

On the construction of cospectral nonisomorphic bipartite graphs

In this article, we construct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices using the notion of partitioned tensor products. This extends the construction of Ji, Gong, and Wang [9]. Our proof of the cospectrality of adjacency matrices simplifies the proof of the bipartite case of Godsil and McKay's construction [4], and shows that the corresponding normalized Laplacian matrices are also cospectral. We partially characterize the isomorphism in Godsil and McKay's construction, and generalize Ji et al.'s characterization of the isomorphism to biregular bipartite graphs. The essential idea in characterizing the isomorphism uses Hammack's cancellation law as opposed to Hall's marriage theorem used by Ji et al.

Measurable Hall’s theorem for actions of abelian groups

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups and measurable sets which are suitably equidistributed with respect to the action, if they are are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

Minimum [formula omitted]-critical bipartite graphs

We study the problem of Minimum k-Critical Bipartite Graph of order (n,m) - MkCBG-(n,m): to find a bipartite G=(U,V;E), with |U|=n, |V|=m, and n>m>1, which is k-critical bipartite, and the tuple (|E|,ΔU,ΔV), where ΔU and ΔV denote the maximum degree in U and V, respectively, is lexicographically minimum over all such graphs. G is k-critical bipartite if deleting at most k=n−m vertices from U creates G′ that has a complete matching, i.e., a matching of size m. We show that, if m(n−m+1)∕n is an integer, then a solution of the MkCBG-(n,m) problem can be found efficiently among (a,b)-regular bipartite graphs of order (n,m), with a=m(n−m+1)∕n, and b=n−m+1. If a=m−1, then all (a,b)-regular bipartite graphs of order (n,m) are k-critical bipartite. For a<m−1, it is not the case. We characterize the values of n, m, a, and b that admit an (a,b)-regular bipartite graph of order (n,m), with b=n−m+1, and give a simple construction that creates such a k-critical bipartite graph whenever possible. Our techniques are based on Hall’s marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.

Constructing cospectral bipartite graphs

Given a bipartite graph G=(V,E) with bipartition V=A∪B, where |A|=|B|. We present a new construction of pairs of cospectral bipartite graphs by “rotating” G in two different ways. More precisely, let l and k be two positive integers with l≠k. Take l copies of A (resp. B) and k copies of B (resp. A), say A1,A2,…,Al and B1,B2,…,Bk (resp. B1′,B2′,…,Bl′ and A1′,A2′,…,Ak′). Let Γ (resp. Γ′) be the graph with vertex set V(Γ)=A1∪A2∪⋯∪Al∪B1∪B2∪⋯∪Bk (resp. V(Γ′)=B1′∪B2′∪⋯∪Bl′∪A1′∪A2′∪⋯∪Ak′), such that the vertices of Γ (resp. Γ′) are adjacent if and only if they are adjacent in G. We show that Γ and Γ′ are cospectral with respect to the adjacency matrix, as well as the normalized Laplacian matrix. Moreover, we give a simple necessary and sufficient condition for Γ and Γ′ to be non-isomorphic. The main ingredient of the proof involves a use of Hall’s Marriage Theorem.

Unbiased Version of Hall’s Marriage Theorem in Matrix Form

Inspired by an old result by Georg Frobenius, we show that the unbiased version of Hall’s marriage theorem is more transparent when reformulated in the language of matrices. At the same time, we obtain a more general statement applicable to bipartite graphs whose parts need not have the same size.

A proof of the Erdős–Sands–Sauer–Woodrow conjecture

A very nice result of Bárány and Lehel asserts that every finite subset X or Rd can be covered by h(d)X-boxes (i.e. each box has two antipodal points in X). As shown by Gyárfás and Pálvőlgyi this result would follow from the following conjecture: If a tournament admits a partition of its arc set into k quasi-orders, then its domination number is bounded in terms of k. This question is in turn implied by the Erdős–Sands–Sauer–Woodrow conjecture: If the arcs of a tournament T are coloured with k colour's, there is a set X of at most g(k) vertices such that for every vertex v of T, there is a monochromatic path from X to v. We give a short proof of this statement. We moreover show that the general Sands–Sauer–Woodrow conjecture (which as a special case implies the stable marriage theorem) is valid for directed graphs with bounded stability number. This conjecture remains however open.

Graphs, friends and acquaintances

As is well known, a graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. Therefore, it is not surprising that many of the first results concerning graphs made reference to relationships between people or groups of people. In this article, we comment on four results of this kind, which are related to various general theories on graphs and their applications: the Handshake lemma (related to graph colorings and Boolean algebra), a lemma on known and unknown people at a cocktail party (to Ramsey theory), a theorem on friends in common (to distance-regularity and coding theory), and Hall's Marriage theorem (to the theory of networks). These four areas of graph theory, often with problems which are easy to state but difficult to solve, are extensively developed and currently give rise to much research work. As examples of representative problems and results of these areas, which are discussed in this paper, we may cite the following: the Four Colors Theorem (4CTC), the Ramsey numbers, problems of the existence of distance-regular graphs and completely regular codes, and finally the study of topological proprieties of interconnection networks.

Cube term blockers without finiteness

We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall’s Marriage Theorem to show that an idempotent variety $${\mathcal{V}}$$ of finite signature whose fundamental operations have arities n 1, . . . , n k, has a d-dimensional cube term for some d if and only if it has one of dimension $${1 + \sum_{i=1}^{k} (n_{i} - 1)}$$ . This upper bound on the dimension of a minimal-dimension cube term for $${\mathcal{V}}$$ is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2- element semilattice. We prove that the Maltsev condition “existence of a cube term” is join prime in the lattice of idempotent Maltsev conditions.

Bargaining Frictions in Trading Networks

AbstractIn the canonical model of frictionless markets, arbitrage is usually taken to force all trades of homogeneous goods to occur at essentially the same price. In the real world, however, arbitrage possibilities are often severely restricted and this may lead to substantial price heterogeneity. Here we focus on frictions that can be modeled as the bargaining constraints induced by an incomplete trading network. In this context, the interplay among the architecture of the trading network, the buyers’ valuations, and the sellers’ costs shapes the effective arbitrage possibilities of the economy. We characterize the configurations that, at an intertemporal bargaining equilibrium, lead to a uniform price. Conceptually, this characterization involves studying how the network positions and valuations/costs of any given set of buyers and sellers affect their collective bargaining power relative to a notional or benchmark situation in which the connectivity is complete. Mathematically, the characterizing conditions can be understood as price-based counterparts of those identified by the celebrated Marriage Theorem in matching theory.

Skills management in the optimization of aircraft maintenance processes

The aircraft maintenance processes play an important role in a safe operation of an aircraft. Maintenance services organizations take responsibility for the maintenance process and approve the airworthiness of an aircraft after undertaking the maintenance activities. International law determines the quality of aircraft maintenance processes by setting requirements concerning, among other, a quality management system, a safety management system and operators’ competences. As a consequence of the rising number of aircraft in operation, the volume of maintenance activities grows. However, the customers increasingly pose requirements concerning the minimization of the maintenance service lead time. In order to remain competitive, the maintenance service organizations have to reduce the lead time of their services. However, this objective in not easy to attain, since the complexity of aircraft maintenance operations require specific skills and pose a number of organisational and technical constraints to be respected during the maintenance process. In the paper, a mathematical programming model is developed in order to help decision makers in managing the operators’ skills during the operators assignment to the activities to be performed. In particular, the Hall’s marriage theorem is used to formalise complex restrictions of operators assignment to maintenance activities. The objective of the optimization problem is to minimize total makespan time. The model is applied to a case study.

Two-Sided, Unbiased Version of Hall's Marriage Theorem

The standard conditions in Hall's perfect matching theorem for a bipartite graph G require that all subsets from one side of G are expanding. The unbiased extension identifies mixtures of subsets from both sides such that their expansions imply the standard conditions—hence a perfect matching.

Every Graph $G$ is Hall $\Delta(G)$-Extendible

In the context of list coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $k$-extendible for some $k \geq \chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \geq \chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.

Generalizations of Marriage Theorem for Degree Factors

Let G be a bipartite graph with bipartition (A, B). We give new criteria for a bipartite graph to have an f -factor, a (g, f)-factor and other factors together with some applications of these criteria. These criteria can be considered as direct generalizations of Hall's marriage theorem. Among some results, we prove that for a function $$h: A\cup B \rightarrow \{0,1,2, \ldots \}$$h:AźBź{0,1,2,ź}, G has a factor F such that $$\deg _F(x)=h(x)$$degF(x)=h(x) for $$x\in A$$xźA and $$\deg _H(y) \le h(y)$$degH(y)≤h(y) for $$y\in B$$yźB if and only if $$h(X) \le \sum _{x\in N_G(X)}\min \{h(x), e_G(x,X)\}$$h(X)≤źxźNG(X)min{h(x),eG(x,X)} for all $$X\subseteq A$$X⊆A.

Completing Partial Proper Colorings using Hall's Condition

In the context of list-coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph $G$ with list assignment $L$ satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $m$-completable for some $m \geq \chi(G)$ if every partial proper $m$-coloring of $G$ whose corresponding list assignment is Hall can be extended to a proper coloring of $G$. In 2011, Bobga et al. posed the following questions: (1) Are there examples of graphs that are Hall $m$-completable, but not Hall $(m+1)$-completable for some $m \geq 3$? (2) If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-completable? This paper establishes that for every $m \geq 3$, there exists a graph that is Hall $m$-completable but not Hall $(m+1)$-completable and also that every bipartite planar graph $G$ is Hall $\Delta(G)$-completable.

A geometric Hall-type theorem

We introduce a geometric generalization of Hall's marriage theorem. For any family $F = \{X_1, \dots, X_m\}$ of finite sets in $\mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_i\in X_i$ for every $1\leq i \leq m$ in such a way that the points $\{x_1,...,x_m\}\subset \mathbb{R}^d$ are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell's celebrated generalization of Hall's theorem.

Systems of distant representatives in Euclidean space

Given a finite family of sets, Hall's classical marriage theorem provides a necessary and sufficient condition for the existence of a system of distinct representatives for the sets in the family. Here we extend this result to a geometric setting: given a finite family of objects in the Euclidean space (e.g., convex bodies), we provide a sufficient condition for the existence of a system of distinct representatives for the objects that are also distant from each other. For a wide variety of geometric objects, this sufficient condition is also necessary in an asymptotic sense (i.e., apart from constant factors, the inequalities are the best possible). Our methods are constructive and lead to efficient algorithms for computing such representatives.