Perfect Matchings Research Articles

Two counterparts of the TFK formula for cylinder graphs

The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph — due independently to Temperley and Fisher, and Kasteleyn, who found the TFK formula — consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In this paper we present a new method, which relies on the Cauchy-Binet theorem, and use it to solve the two dimensional dimer problem for cylinder graphs on the square and on the hexagonal lattice.We provide explicit product formulas for these two kinds of cylinder graphs. One advantage of our formula for the square lattice compared to the TFK formula is that ours has a linear number of factors, while the number of factors in the former is quadratic. Our result for the hexagonal lattice yields a formula for the number of periodic stepped surfaces that fit in an infinite tube of given cross-section. This can be regarded as a counterpart of MacMahon's boxed plane partition theorem.

Perfect matchings, Hamiltonian cycles and edge-colourings in a class of cubic graphs

A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and H\"{a}ggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if $G$ is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic.

Erdős matching conjecture for almost perfect matchings

In 1965 Erdős asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size s+1? In this note, we improve upon a recent result of Frankl and resolve this problem for s>101k3 and (s+1)k⩽n<(s+1)(k+1100k).

The maximal spectral radius of the uniform unicyclic hypergraphs with perfect matchings

Let U(n, k) and ?(n, k) be respectively the sets of the k-uniform connected linear and nonlinear unicyclic hypergraphs having perfect matchings with n vertices, where n ? k(k?1) and k ? 3. By using some techniques of transformations and constructing the incidence matrices for the hypergraphs considered, we get the hypergraphs with the maximal spectral radii among three kinds of hypergraphs, namely U(n, k) with n = 2k(k ? 1) and n ? 9k(k ? 1), ?(n, k) with n ? k(k ? 1), and U(n, k) ? ?(n, k) with n ? 2k(k ? 1), where k ? 3.

Rainbow spanning structures in graph and hypergraph systems

Abstract We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection $\mathit {\mathbf {G}}=\{G_1, G_2,\ldots , G_{m}\}$ of not necessarily distinct k-graphs on the same vertex set $[n]$ , a (sub)graph H on $[n]$ is rainbow if there exists an injection $\varphi : E(H)\rightarrow [m]$ , such that $e\in E(G_{\varphi (e)})$ for each $e\in E(H)$ . Note that if $|E(H)|=m$ , then $\varphi $ is a bijection, and thus H contains exactly one edge from each $G_i$ . Our main results focus on rainbow clique-factors in (hyper)graph systems with minimum d-degree conditions. Specifically, we establish the following: (1) A rainbow analogue of an asymptotical version of the Hajnal–Szemerédi theorem, namely, if $t\mid n$ and $\delta (G_i)\geq (1-\frac {1}{t}+\varepsilon )n$ for each $i\in [\frac {n}{t}\binom {t}{2}]$ , then $\mathit {\mathbf {G}}$ contains a rainbow $K_t$ -factor; (2) Essentially, a minimum d-degree condition forcing a perfect matching in a k-graph also forces rainbow perfect matchings in k-graph systems for $d\in [k-1]$ . The degree assumptions in both results are asymptotically best possible (although the minimum d-degree condition forcing a perfect matching in a k-graph is in general unknown). For (1), we also discuss two directed versions and a multipartite version. Finally, to establish these results, we in fact provide a general framework to attack this type of problem, which reduces it to subproblems with finitely many colors.

Extremal quasi-unicyclic graphs with respect to the general multiplicative Zagreb indices

The general multiplicative Zagreb indices are modified versions of the famous Zagreb indices, which can help to predict the various activities and properties of organic compounds. In this article, the minimum and maximum general multiplicative Zagreb indices of quasi-unicyclic graphs with given order, with fixed number of pendant vertices and with perfect matchings are investigated. Furthermore, the corresponding extremal quasi-unicyclic graphs are identified.

Symmetric property and the bijection between perfect matchings and sub-hypercubes of enhanced hypercubes

An ( n , k ) -enhanced hypercube Q n , k ( 1 ≤ k ≤ n − 1 ) is an extended structure of an n -dimensional hypercube Q n . Q n , k possesses many properties that are superior to those of Q n including the diameter, fault diameter, and connectivity. For 1 ≤ i ≤ n , let P M i = { ( u , u i ) | u ∈ V ( Q n , k ) } , P M = { ( u , u ̄ ) | u ∈ V ( Q n , k ) } , where u = u 1 u 2 ⋯ u n , u ̄ = u 1 ⋯ u k − 1 u ̄ k u ̄ k + 1 ⋯ u ̄ n , and u i = u 1 u 2 ⋯ u i − 1 u ̄ i u i + 1 ⋯ u n , u i = 0 or 1, i = 1 , 2 , … , n . In this note, we prove that for any perfect matching M of Q n , k , if n ≥ 4 , 2 ≤ k ≤ n − 1 and k ≠ n − 2 or n = 3 , k = 2 , then Q n , k − M is isomorphic to Q n if and only if M = P M or P M i where i ∈ { k , k + 1 , … , n } , which extends Lü and Wu’s result in [Discrete Applied Mathematics, 279 (2020) 192–194]. Based on this result, we show that Q n , k with 3 ≤ k + 1 ≤ n is not edge-transitive.

Complete forcing numbers of catacondensed phenylene systems

Combining ?forcing? and ?global? idea, Xu et al. proposed the concepts: complete forcing set and complete forcing number of perfect matchings of graph. In this paper, we give explicit formulae for the complete forcing numbers of phenylene chains and catacondensed phenylene systems, respectively. Moreover, we present an algorithm to find the minimum complete forcing sets of these graphs.

COMBINATORIAL PROPERTIES FOR A CLASS OF SIMPLICIAL COMPLEXES EXTENDED FROM PSEUDO-FRACTAL SCALE-FREE WEB

Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a graph product, which is an extension of the pseudo-fractal scale-free web. We determine explicitly the independence number, the domination number, and the chromatic number. Moreover, we derive closed-form expressions for the number of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning trees, as well as the number of perfect matchings for some particular cases.

Certain structural properties for Cayley regularity graphs of semigroups and their theoretical applications

&lt;abstract&gt;&lt;p&gt;An element $ x $ in a semigroup is said to be regular if there exists an element $ y $ in the semigroup such that $ x = xyx $. The element $ y $ is said to be a regular part of $ x $. Define the Cayley regularity graph of a semigroup $ S $ to be a digraph with vertex set $ S $ and arc set containing all ordered pairs $ (x, y) $ such that $ y $ is a regular part of $ x $. In this paper, certain classes of Cayley regularity graphs such as complete digraphs, connected digraphs and equivalence digraphs are investigated. Furthermore, structural properties of the Cayley regularity graphs are theoretically applied to study perfect matchings of other algebraic graphs.&lt;/p&gt;&lt;/abstract&gt;

Smoothed counting of 0–1 points in polyhedra

AbstractGiven a system of linear equations in an ‐vector of 0–1 variables, we compute the expectation of , where is a vector of independent Bernoulli random variables and are constants. The algorithm runs in quasi‐polynomial time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising model with a sufficiently strong external field. We discuss applications to perfect matchings in hypergraphs and randomized rounding in discrete optimization.

Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching

Let G be a bridgeless cubic graph. The Berge–Fulkerson Conjecture (1970s) states that G admits a list of six perfect matchings such that each edge of G belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan–Raspaud Conjecture (1994), which states that G admits three perfect matchings such that every edge of G belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that G admits two perfect matchings whose deletion yields a bipartite subgraph of G. It can be shown that given an arbitrary perfect matching of G, it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan–Raspaud and the Berge–Fulkerson conjectures, respectively. In this paper, we show that given any 1+-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge e of G, there always exists a perfect matching M of G containing e such that G∖(F∪M) is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in G, there exists a perfect matching of G containing at least one edge of each circuit in this collection.

Distance spectral radius of the complements of trees with fixed parameters

Let G be a simple connected graph and Gc be the complement of G. Denote by dG(u,v) the distance between the vertices u and v which is the length of the shortest path between u and v in G. Let D(G)=(dG(u,v))n×n be the distance matrix of G and λ1(D(G)) denote the distance spectral radius of G. In this paper, we characterize the unique graphs with maximum distance spectral radius among all the complements of trees of order n with fixed maximum degree, pendent vertices, diameter and perfect matchings, respectively.

Гафниан двухпараметрических матриц

The concept of the hafnian first appeared in the works on&#x0D; quantum field theory by E. R. Caianiello. However, it also has&#x0D; an important combinatorial property: the hafnian of the adjacency&#x0D; matrix of an undirected weighted graph is equal to the&#x0D; total sum of the weights of perfect matchings in this graph.&#x0D; In general, the use of the hafnian is limited by the complexity&#x0D; of its computation. In this paper, we present a method for the&#x0D; exact calculation of the hafnian of two-parameter matrices.&#x0D; In terms of graphs, we count the total sum of the weights of&#x0D; perfect matchings in graphs whose edge weights take only&#x0D; two values. This method is based on the formula expressing&#x0D; the hafnian of a sum of two matrices through the product of&#x0D; the hafnians of their submatrices. The necessary condition&#x0D; for the application of this method is the possibility to count&#x0D; the number of k-edge matchings in some graphs. We consider&#x0D; a special case in detail using a Toeplitz matrix as the&#x0D; two-parameter matrix. As an example, we propose a new interpretation&#x0D; of some of the sequences from the On-Line Encyclopedia&#x0D; of Integer Sequences and then provide new analytical&#x0D; formulas to count the number of certain linear chord&#x0D; diagrams.

Reliability analysis of the generalized balanced hypercube

With the rapid and continuous development of network technology, a multiprocessor system, consisting of multiple processors and communication links, plays an significant role in big data era. The topology of a multiprocessor system is a basic architecture of great influence on the network performance. Multiprocessor systems should have some good performances such as low communication costs, high network reliability, and friendly resource sharing. Therefore, designing a high-performance network architecture is of great significance to multiprocessor systems. In this paper, we propose a more general class of balanced hypercubes with two more general edge-disjoint sets of perfect matchings as cross edges. Then we explore some basic properties of GBHn. Furthermore, we show the connectivity, super connectivity, extra connectivity and (extra) diagnosability of GBHn, respectively.

Graft Analogue of general Kotzig–Lovász decomposition

Canonical decompositions, such as the Gallai–Edmonds and Dulmage–Mendelsohn decompositions, are a series of structure theorems that form the foundation of matching theory. The classical Kotzig–Lovász decomposition is a canonical decomposition applicable to a special class of graphs with perfect matchings, called factor-connected graphs, and is famous for its contribution to the studies of perfect matching polytopes and lattices. Recently, this decomposition has been generalized for general graphs with perfect matchings; this generalized decomposition is called the general Kotzig–Lovász decomposition . In fact, this generalized decomposition can be presented as a component of a more composite canonical decomposition called the Basilica decomposition . As such, the general Kotzig–Lovász decomposition has contributed to the derivation of various new results in matching theory, such as an alternative proof of the tight cut lemma and a characterization of maximal barriers in general graphs. Joins in grafts, also known as T -joins in graphs, is a classical concept in combinatorics and is a generalization of perfect matchings in terms of parity. More precisely, minimum joins and grafts are generalizations of perfect matchings and graphs with perfect matchings, respectively. Under the analogy between matchings and joins, analogues of the canonical decompositions for grafts are expected to be strong and fundamental tools for studying joins. In this paper, we provide a generalization of the general Kotzig–Lovász decomposition for arbitrary grafts. Our result also contains Sebö’s generalization of the classical Kotzig–Lovász decomposition for factor-connected grafts. From our results in this paper, a generalization of the Dulmage–Mendelsohn decomposition, which is originally a classical canonical decomposition for bipartite graphs, has been obtained for bipartite grafts. This paper is the first of a series of papers that establish a generalization of the Basilica decomposition for grafts.

Computational and topological properties of neural networks by means of graph-theoretic parameters

A neural network is a computer system modeled on the nerve tissue and nervous system. In this sense, neural networks refer to systems of neurons, either organic or artificial in nature. Neural networks have diverse applications in computer graphics, artificial intelligence (AI), machine & deep learning, chemistry, material science, among others. Research on structural properties and their classification of favorable or unfavorable for neural networks has been started recently. Employing tools from mathematics and specifically graph theory has been a key research direction in this area. Different graph-theoretic parameters have potential applications in studying topological properties of neural networks. In this paper for the first time, we study some nondeterministic polynomial-time (NP)-complete and NP-hard problems on various neural networks by considering their graphs. We consider classes of 3- and 4-layered probabilistic neural networks (PNNs), cellular neural networks (CNNs), and Tickysym spiking neural networks (TSNNs). In order to study their topological properties, we study their structures of maximal cliques, minimal colorings, maximal independent sets, maximal and perfect matchings, and minimum dominating sets. Computational results on the clique number, chromatic number, independence number, matching ratio and the domination number are reported. For instance, the clique number of the probabilistic neural networks is 2, the Tickysym spiking neural network is 3, whereas, for the cellular neural network, it is 4. These numerical results quantify the incoming/outgoing traffic in these architectures. Similarly, the independence number of 25-nodal 3-layered probabilistic neural network is 15, whereas, the independence number of 25-nodal cellular neural network is just 9. Moreover, the asymptotic matching ratio for PNNs (resp. CNNs) is 12 (resp. 18), whereas, it is 16 for the case of TSNNs. This gives probabilistic neural networks a significant priority in controllability over cellular neural networks. Some open problems are proposed in the end to further this research area.

From One to Many Rainbow Hamiltonian Cycles

Given a graph G and a family \(\mathcal {G} = \{G_1,\ldots ,G_n\}\) of subgraphs of G, a transversal of \(\mathcal {G}\) is a pair \((T,\phi )\) such that \(T \subseteq E(G)\) and \(\phi : T \rightarrow [n]\) is a bijection satisfying \(e \in G_{\phi (e)}\) for each \(e \in T\). We call a transversal \((T, \phi )\) Hamiltonian if T corresponds to the edge set of a Hamiltonian cycle in G. We show that, under certain conditions on the maximum degree of G and the minimum degrees of the \(G_i \in \mathcal {G}\), for every \(\mathcal {G}\) which contains a Hamiltonian transversal, the number of Hamiltonian transversals contained in \(\mathcal {G}\) is bounded below by a function of G’s maximum degree. This generalizes a theorem of Thomassen stating that, for \(m \ge 300\), no m-regular graph is uniquely Hamiltonian. We also extend Joos and Kim’s recent result that, if \(G = K_{n}\) and each \(G_i \in \mathcal {G}\) has minimum degree at least n/2, then \(\mathcal {G}\) has a Hamiltonian transversal: we show that, in this setting, \(\mathcal {G}\) has factorially many Hamiltonian transversals. Finally, we prove analogues of both of these theorems for transversals which form perfect matchings in G.

Kasteleyn cokernels and perfect matchings on planar bipartite graphs

The determinant method of Kasteleyn gives a method of computing the number of perfect matchings of a planar bipartite graph. In addition, results of Bernardi exhibit a bijection between spanning trees of a planar bipartite graph and elements of its Jacobian. In this paper, we explore an adaptation of Bernardi’s results, providing a simply transitive group action of the Kasteleyn cokernel of a planar bipartite graph on its set of perfect matchings, when the planar bipartite graph in question is of the form $$G^+$$ , as defined by Kenyon, Propp and Wilson.

Certification of Gaussian Boson Sampling via graphs feature vectors and kernels

Gaussian Boson Sampling (GBS) is a non-universal model for quantum computing inspired by the original formulation of the Boson Sampling (BS) problem. Nowadays, it represents a paradigmatic quantum platform to reach the quantum advantage regime in a specific computational model. Indeed, thanks to the implementation in photonics-based processors, the latest GBS experiments have reached a level of complexity where the quantum apparatus has solved the task faster than currently up-to-date classical strategies. In addition, recent studies have identified possible applications beyond the inherent sampling task. In particular, a direct connection between photon counting of a genuine GBS device and the number of perfect matchings in a graph has been established. In this work, we propose to exploit such a connection to benchmark GBS experiments. We interpret the properties of the feature vectors of the graph encoded in the device as a signature of correct sampling from the true input state. Within this framework, two approaches are presented. The first method exploits the distributions of graph feature vectors and classification via neural networks. The second approach investigates the distributions of graph kernels. Our results provide a novel approach to the actual need for tailored algorithms to benchmark large-scale Gaussian Boson Samplers.