Perfect Matchings Research Articles

Fractional matching preclusion for radix triangular mesh

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we study the fractional matching preclusion number and the fractional strong matching preclusion number for the radix triangular mesh [Formula: see text], and all the optimal fractional matching preclusion sets and fractional strong matching preclusion sets of these graphs are categorized.

Fractional matching preclusion of the restricted HL-graphs

The fractional matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings, and the fractional strong matching preclusion number of a graph is the minimum number of edges and/or vertices whose deletion leaves a resulting graph with no fractional perfect matchings. In this paper, we determine these two numbers for the restricted HL-graphs.

The complexity of perfect matchings and packings in dense hypergraphs

Given two k-graphs H and F, a perfect F-packing in H is a collection of vertex-disjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F-packing is simply a perfect matching. For a given fixed F, it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F-packing is NP-complete. Indeed, if k≥3, the corresponding problem for perfect matchings is NP-complete [17,7] whilst if k=2 the problem is NP-complete in the case when F has a component consisting of at least 3 vertices [14].In this paper we give a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect F-packings is polynomial time solvable. We then give three applications of this tool: (i) Given 1≤ℓ≤k−1, we give a minimum ℓ-degree condition for which it is polynomial time solvable to determine whether a k-graph satisfying this condition has a perfect matching; (ii) Given any graph F we give a minimum degree condition for which it is polynomial time solvable to determine whether a graph satisfying this condition has a perfect F-packing; (iii) We also prove a similar result for perfect K-packings in k-graphs where K is a k-partite k-graph.For a range of values of ℓ,k (i) resolves a conjecture of Keevash, Knox and Mycroft [20]; (ii) answers a question of Yuster [47] in the negative; whilst (iii) generalises a result of Keevash, Knox and Mycroft [20]. In many cases our results are best possible in the sense that lowering the minimum degree condition means that the corresponding decision problem becomes NP-complete.

Computing permanents of complex diagonally dominant matrices and tensors

We prove that for any λ > 1, fixed in advance, the permanent of an n × n complex matrix, where the absolute value of each diagonal entry is at least λ times bigger than the sum of the absolute values of all other entries in the same row, can be approximated within any relative error 0 < ϵ < 1 in quasi-polynomial nO(lnn-lnϵ) time. We extend this result to multidimensional permanents of tensors and apply it to weighted counting of perfect matchings in hypergraphs.

A short proof of the toughness of Delaunay triangulations

We present a self-contained short proof of the seminal result of Dillencourt (SoCG 1987 and DCG 1990) that Delaunay triangulations, of planar point sets in general position, are 1-tough. An important implication of this result is that Delaunay triangulations have perfect matchings. Another implication of our result is a proof of the conjecture of Aichholzer et al. (2010) that at least n points are required to block any n-vertex Delaunay triangulation.

Anti-Ramsey Problems in Complete Bipartite Graphs for t Edge-Disjoint Rainbow Spanning Subgraphs: Cycles and Matchings

A subgraph H of an edge-colored graph G is rainbow if all of its edges have different colors. The anti-Ramsey number is the maximum number of colors in an edge-coloring of G with no rainbow copy of H. Originally a complete graph was considered as G. In this paper, we consider a complete bipartite graph as the host graph and discuss some results for the graph H being hamiltonian cycle and perfect matching. Let $$c(K_{p,p},t)$$ and $$m(K_{p,p},t)$$ be the maximum number of colors in an edge-coloring of the complete bipartite graph $$K_{p,p}$$ not having t edge-disjoint rainbow hamiltonian cycles and perfect matchings, respectively. We prove that $$c(K_{p,p},t)=p^2-p+t$$ for $$t \ge 2$$ , $$p\ge 4t-1$$ and $$m(K_{p,p},t)=p^2-2p+t+1$$ for $$t \ge $$ 2, $$p\ge 2t+8$$ .

A Faster Algorithm for Cuckoo Insertion and Bipartite Matching in Large Graphs

Hash tables are ubiquitous in computer science for efficient access to large datasets. However, there is always a need for approaches that offer compact memory utilisation without substantial degradation of lookup performance. Cuckoo hashing is an efficient technique of creating hash tables with high space utilisation and offer a guaranteed constant access time. We are given n locations and m items. Each item has to be placed in one of the $$k\ge 2$$ locations chosen by k random hash functions. By allowing more than one choice for a single item, cuckoo hashing resembles multiple choice allocations schemes. In addition it supports dynamically changing the location of an item among its possible locations. We propose and analyse an insertion algorithm for cuckoo hashing that runs in linear time with high probability and in expectation. Previous work on total allocation time has analysed breadth first search, and it was shown to be linear only in expectation. Our algorithm finds an assignment (with probability 1) whenever it exists. In contrast, the other known insertion method, known as random walk insertion, may run indefinitely even for a solvable instance. We also present experimental results comparing the performance of our algorithm with the random walk method, also for the case when each location can hold more than one item. As a corollary we obtain a linear time algorithm (with high probability and in expectation) for finding perfect matchings in a special class of sparse random bipartite graphs. We support this by performing experiments on a real world large dataset for finding maximum matchings in general large bipartite graphs. We report an order of magnitude improvement in the running time as compared to the Hopkraft–Karp matching algorithm.

A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer

We introduce new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops. Our compact formulas for the hafnian and loop hafnian of n × n complex matrices run in O ( n 3 2 n /2 ) time, are embarrassingly parallelizable and, to the best of our knowledge, are the fastest exact algorithms to compute these quantities. Despite our highly optimized algorithm, numerical benchmarks on the Titan supercomputer with matrices up to size 56 × 56 indicate that one would require the 288,000 CPUs of this machine for about 6 weeks to compute the hafnian of a 100 × 100 matrix.

Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers

Abstract In this paper, we consider the relationships between the numbers of perfect matchings (1-factors) of bipartite graphs and Pell, Mersenne and Perrin Numbers. Then we give some Maple procedures in order to calculate the numbers of perfect matchings of these bipartite graphs.

Combinatorial Cluster Expansion Formulas from Triangulated Surfaces

We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of maximal independent sets of angles. Our formula simplifies the cluster expansion formula given by Musiker, Schiffler and Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between maximal independent sets of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential.&#x0D;

A note on generalized matching preclusion in bipartite graphs

For a graph with an even number of vertices, the matching preclusion number is the minimum number of edges whose deletion results in a graph with no perfect matchings. The conditional matching preclusion number, introduced as an extension of the matching preclusion number, has the additional requirement that the resulting graph have no isolated vertices. In bipartite graphs, the edge sets that achieve these minimum numbers can be construed in terms of obstruction sets of vertices. From this perspective, the generalized matching preclusion problem was developed; the matching preclusion number of order a is the minimum number of edges whose deletion results in a graph with no perfect matchings and no obstruction sets with fewer than a vertices. In this paper, we extend known sufficient conditions regarding the matching preclusion and conditional matching preclusion numbers (orders 1 and 2, respectively) to give sufficient conditions for the matching preclusion problem of order a, for any positive integer a.

Permanental generating functions and sequential importance sampling

We introduce techniques for deriving closed form generating functions for enumerating permutations with restricted positions keeping track of various statistics. The method involves evaluating permanents with variables as entries. These are applied to determine the sample size required for a novel sequential importance sampling algorithm for generating random perfect matchings in classes of bipartite graphs.

Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we study the del Pezzo 2 (dP$_2$) quiver and its associated brane tiling which arise in theoretical physics. Specifically, we prove explicit formulas for all cluster variables generated by toric mutation sequences of the dP$_2$ quiver. Moreover, we associate a subgraph of the dP$_2$ brane tiling to each toric cluster variable such that the sum of weighted perfect matchings of the subgraph equals the Laurent polynomial of the cluster variable.

Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions

We study combinatorial properties of polytopes realizable in the Lobachevsky space $$\mathbb{L}^{3}$$ as polytopes of finite volume with right dihedral angles. On the basis of E. M. An-dreev’s theorem we prove that cutting off ideal vertices of right-angled polytopes defines a one-to-one correspondence with strongly cyclically four-edge-connected polytopes different from the cube and the pentagonal prism. We show that any polytope of the latter family can be obtained by cutting off a matching of a polytope from the same family or of the cube with at most two nonadjacent orthogonal edges cut, in such a way that each quadrangle results from cutting off an edge. We refine D. Barnette’s construction of this family of polytopes and present its application to right-angled polytopes. We refine the known construction of ideal right-angled polytopes using edge twists and describe its connection with D. Barnette’s construction via perfect matchings. We make a conjecture on the behavior of the volume under operations and give arguments to support it.

The Wiener index of bicyclic graphs with perfect matchings

The Wiener index of a connected graph is the sum of distances between all pairs of distinct vertices, which is the oldest topological index related to molecular branching. In this paper we give a formula applied to compute the Wiener index of Θ-graphs. With the formula we determine the graphs having the minimum or maximum Wiener index among all Θ-graphs with a fixed order and characterize the graphs having the minimum Wiener index among all bicyclic graphs with perfect matchings.

Franck-Condon factors by counting perfect matchings of graphs with loops.

We show that the Franck-Condon Factor (FCF) associated with a transition between initial and final vibrational states in two different potential energy surfaces, having N and M vibrational quanta, respectively, is equivalent to calculating the number of perfect matchings of a weighted graph with loops that has P = N + M vertices. This last quantity is the loop hafnian of the (symmetric) adjacency matrix of the graph which can be calculated in O(P32P/2) steps. In the limit of small numbers of vibrational quanta per normal mode, our loop hafnian formula significantly improves the speed at which FCFs can be calculated. Our results more generally apply to the calculation of the matrix elements of a bosonic Gaussian unitary between two multimode Fock states having N and M photons in total and provide a useful link between certain calculations of quantum chemistry, quantum optics, and graph theory.

Weighted counting of solutions to sparse systems of equations

AbstractGiven complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ &gt; 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β &gt; 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

Complete Forcing Numbers of Spiro Hexagonal Systems

A forcing set of a perfect matching M of a graph G is a subset of M contained in no other perfect matchings of G. A global forcing set of G, introduced by Vukičević et al., is a subset of E(G) on which there are distinct restrictions of any two different perfect matchings of G. Combining the above“forcing” and “global” ideas, Xu et al. [Journal of Combinatorial Optimization 29, no. 4 (2015): 803–14] introduced a complete forcing set of G defined as a subset of E(G) on which the restriction of any perfect matching M of G is a forcing set of M. The minimum cardinality of complete forcing set is the complete forcing number of G. In this paper, we give the explicit expressions for the complete forcing numbers of several classes of spiro hexagonal systems. Moreover, we also discuss some propositions of spiro cata-condensed hexagonal system with perfect matching related to the concept of forcing numbers.

Decompositions into spanning rainbow structures

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive research ever since. Euler posed a problem equivalent to finding properly $n$-edge-coloured complete bipartite graphs $K_{n,n}$ which can be decomposed into rainbow perfect matchings. While there are proper edge-colourings of $K_{n,n}$ without even a single rainbow perfect matching, the theme of this paper is to show that with some very weak additional constraints one can find many disjoint rainbow perfect matchings. In particular, we prove that if some fraction of the colour classes have at most $(1-o(1)) n$ edges then one can nearly-decompose the edges of $K_{n,n}$ into edge-disjoint perfect rainbow matchings. As an application of this, we establish in a very strong form a conjecture of Akbari and Alipour and asymptotically prove a conjecture of Barat and Nagy. Both these conjectures concern rainbow perfect matchings in edge-colourings of $K_{n,n}$ with quadratically many colours. Using our techniques, we also prove a number of results on near-decompositions of graphs into other rainbow structures like Hamiltonian cycles and spanning trees. Most notably, we prove that any properly coloured complete graph can be nearly-decomposed into spanning rainbow trees. This asymptotically proves the Brualdi-Hollingsworth and Kaneko-Kano-Suzuki conjectures which predict that a perfect decomposition should exist under the same assumptions.

The hyper-Zagreb index of cacti with perfect matchings

Let G be a simple connected graph. The hyper-Zagreb index is defined as HM(G)=∑uv∈E(G)(dG(u)+dG(v))2. A connected graph G is a cacti if all blocks of G are either edges or cycles. Let ζ(2n,r) be the set of cacti of order 2n with a perfect matching and r cycles. In this paper, we determine sharp upper bounds of the hyper-Zagreb index of cacti among ζ(2n,r) and characterize the corresponding extremal cacti.