A CHARACTERIZATION OF PM-COMPACT CLAW-FREE CUBIC GRAPHS

A graph $G$ with a perfect matching is Pfaffian if it admits an orientation $D$ such that every central cycle $C$ (i.e. $C$ is of even size and $G-V(C)$ has a perfect matching) has an odd number of edges oriented in either direction of the cycle. It is known that the number of perfect matchings of a Pfaffian graph can be computed in polynomial time. In this paper, we show that every embedding of a Pfaffian brace (i.e. 2-extendable bipartite graph) on a surface with a positive genus has face-width at most 3. Further, we study Pfaffian cubic braces and obtain a characterization of Pfaffian polyhex graphs: a polyhex graph is Pfaffian if and only if it is either non-bipartite or isomorphic to the cube, or the Heawood graph, or the Cartesian product $C_k\times K_2$ for even integers $k\ge 6$.

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Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

Special Matrices ◽

10.1515/spma-2018-0028 ◽

2018 ◽

Vol 6 (1) ◽

pp. 343-356

Author(s):

K. Arathi Bhat ◽

G. Sudhakara

Keyword(s):

Chromatic Number ◽

Perfect Matching ◽

Domination Number ◽

Perfect Matchings ◽

Factor Graph ◽

Vertex Set ◽

Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

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Perfect Matching Covers of Cubic Graphs of Oddness 2

The Electronic Journal of Combinatorics ◽

10.37236/7175 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Wuyang Sun ◽

Fan Wang

Keyword(s):

Perfect Matching ◽

Perfect Matchings ◽

Cubic Graphs ◽

Cubic Graph

A perfect matching cover of a graph $G$ is a set of perfect matchings of $G$ such that each edge of $G$ is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph has a perfect matching cover of order at most 5. The Berge Conjecture is largely open and it is even unknown whether a constant integer $c$ does exist such that every bridgeless cubic graph has a perfect matching cover of order at most $c$. In this paper, we show that a bridgeless cubic graph $G$ has a perfect matching cover of order at most 11 if $G$ has a 2-factor in which the number of odd circuits is 2.

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A Note on the Matching Polytope of a Graph

TEMA (São Carlos) ◽

10.5540/tema.2019.020.01.189 ◽

2019 ◽

Vol 20 (1) ◽

pp. 189 ◽

Cited By ~ 1

Author(s):

Nair Maria Maia de Abreu ◽

Liliana Manuela Gaspar Cerveira da Costa ◽

Carlos Henrique Pereira Nascimento ◽

Laura Patuzzi

Keyword(s):

Convex Hull ◽

Minimum Degree ◽

Arbitrary Graph ◽

Matching Polytopes ◽

Matching Polytope

The matching polytope of a graph G, denoted by M(G), is the convex hull of the set of the incidence vectors of the matchings G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G. In this paper, for an arbitrary graph, we prove that the minimum degree of G(M(G)) is equal to the number of edges of G, generalizing a known result for trees. From this, we identify the vertices of the skeleton with the minimum degree and we prove that the union of stars and triangles characterizes regular skeletons of the matching polytopes of graphs.

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On the Number of Perfect Matchings in Random Lifts

Combinatorics Probability Computing ◽

10.1017/s0963548309990654 ◽

2010 ◽

Vol 19 (5-6) ◽

pp. 791-817 ◽

Cited By ~ 8

Author(s):

CATHERINE GREENHILL ◽

SVANTE JANSON ◽

ANDRZEJ RUCIŃSKI

Keyword(s):

Asymptotic Formula ◽

Complete Graph ◽

Pairwise Disjoint ◽

Perfect Matching ◽

Perfect Matchings ◽

Lattice Points ◽

Laplace’S Method ◽

Second Moment ◽

Multiple Dimensions ◽

Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

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On the König deficiency of zero-reducible graphs

Journal of Combinatorial Optimization ◽

10.1007/s10878-019-00466-2 ◽

2019 ◽

Vol 39 (1) ◽

pp. 273-292

Author(s):

Miklós Bartha ◽

Miklós Krész

Keyword(s):

Perfect Matching ◽

Linear Time ◽

Perfect Matchings ◽

Covering Number ◽

Matching Number ◽

Empty Graph ◽

Reduction System ◽

Vertex Covering ◽

Np Complete ◽

The Difference

Abstract A confluent and terminating reduction system is introduced for graphs, which preserves the number of their perfect matchings. A union-find algorithm is presented to carry out reduction in almost linear time. The König property is investigated in the context of reduction by introducing the König deficiency of a graph G as the difference between the vertex covering number and the matching number of G. It is shown that the problem of finding the König deficiency of a graph is NP-complete even if we know that the graph reduces to the empty graph. Finally, the König deficiency of graphs G having a vertex v such that $$G-v$$G-v has a unique perfect matching is studied in connection with reduction.

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Perfect 1-factorizations

Mathematica Slovaca ◽

10.1515/ms-2017-0241 ◽

2019 ◽

Vol 69 (3) ◽

pp. 479-496 ◽

Cited By ~ 2

Author(s):

Alexander Rosa

Keyword(s):

Pairwise Disjoint ◽

Hamiltonian Cycle ◽

Perfect Matching ◽

Complete Graphs ◽

Regular Graphs ◽

Cubic Graphs ◽

Vertex Set ◽

Early Survey

AbstractLetGbe a graph with vertex-setV=V(G) and edge-setE=E(G). A 1-factorofG(also calledperfect matching) is a factor ofGof degree 1, that is, a set of pairwise disjoint edges which partitionsV. A 1-factorizationofGis a partition of its edge-setEinto 1-factors. For a graphGto have a 1-factor, |V(G)| must be even, and for a graphGto admit a 1-factorization,Gmust be regular of degreer, 1 ≤r≤ |V| − 1.One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs.A 1-factorization ofGis said to beperfectif the union of any two of its distinct 1-factors is a Hamiltonian cycle ofG. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

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Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

Combinatorics Probability Computing ◽

10.1017/s096354831300028x ◽

2013 ◽

Vol 22 (5) ◽

pp. 783-799 ◽

Cited By ~ 4

Author(s):

GUILLEM PERARNAU ◽

ORIOL SERRA

Keyword(s):

Bipartite Graph ◽

High Probability ◽

Perfect Matching ◽

Complete Bipartite Graph ◽

Bipartite Graphs ◽

Perfect Matchings ◽

Asymptotic Enumeration ◽

Complete Bipartite Graphs ◽

Edge Colourings ◽

Square Matrix

A perfect matchingMin an edge-coloured complete bipartite graphKn,nis rainbow if no pair of edges inMhave the same colour. We obtain asymptotic enumeration results for the number of rainbow perfect matchings in terms of the maximum number of occurrences of each colour. We also consider two natural models of random edge-colourings ofKn,nand show that if the number of colours is at leastn, then there is with high probability a rainbow perfect matching. This in particular shows that almost every square matrix of ordernin which every entry appearsntimes has a Latin transversal.

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Characterization algorithms for shift radix systems with finiteness property

International Journal of Number Theory ◽

10.1142/s179304211550013x ◽

2014 ◽

Vol 11 (01) ◽

pp. 211-232 ◽

Cited By ~ 1

Author(s):

Mario Weitzer

Keyword(s):

Convex Hull ◽

Convex Polyhedron ◽

The Other ◽

Closed Convex Hull ◽

Connected Component ◽

Complicated Structure ◽

Finiteness Property ◽

A Chain ◽

Interior Points

For d ∈ ℕ and r ∈ ℝd, let τr : ℤd → ℤd, where τr(a) = (a2, …, ad, -⌊ra⌋) for a = (a1, …, ad), denote the (d-dimensional) shift radix system associated with r. τr is said to have the finiteness property if and only if all orbits of τr end up in (0, …, 0); the set of all corresponding r ∈ ℝd is denoted by [Formula: see text], whereas 𝒟d consists of those r ∈ ℝd for which all orbits are eventually periodic. [Formula: see text] has a very complicated structure even for d = 2. In the present paper, two algorithms are presented which allow the characterization of the intersection of [Formula: see text] and any closed convex hull of finitely many interior points of 𝒟d which is completely contained in the interior of 𝒟d. One of the algorithms is used to determine the structure of [Formula: see text] in a region considerably larger than previously possible, and to settle two questions on its topology: It is shown that [Formula: see text] is disconnected and that the largest connected component has non-trivial fundamental group. The other is the first algorithm characterizing [Formula: see text] in a given convex polyhedron which terminates for all inputs. Furthermore, several infinite families of "cutout polygons" are deduced settling the finiteness property for a chain of regions touching the boundary of 𝒟2.

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Enumeration of Architectures With Perfect Matchings

Journal of Mechanical Design ◽

10.1115/1.4036132 ◽

2017 ◽

Vol 139 (5) ◽

Cited By ~ 6

Author(s):

Daniel R. Herber ◽

Tinghao Guo ◽

James T. Allison

Keyword(s):

Design Space ◽

Perfect Matching ◽

Search Algorithm ◽

Perfect Matchings ◽

Solution Technique ◽

Architecture Design ◽

Colored Graph ◽

Data Set ◽

Rich Data ◽

Moderate Scale

In this article, a class of architecture design problems is explored with perfect matchings (PMs). A perfect matching in a graph is a set of edges such that every vertex is present in exactly one edge. The perfect matching approach has many desirable properties such as complete design space coverage. Improving on the pure perfect matching approach, a tree search algorithm is developed that more efficiently covers the same design space. The effect of specific network structure constraints (NSCs) and colored graph isomorphisms on the desired design space is demonstrated. This is accomplished by determining all unique feasible graphs for a select number of architecture problems, explicitly demonstrating the specific challenges of architecture design. With this methodology, it is possible to enumerate all possible architectures for moderate scale-systems, providing both a viable solution technique for certain problems and a rich data set for the development of more capable generative methods and other design studies.

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