scholarly journals The 2-Factor Polynomial Detects Even Perfect Matchings

10.37236/9214 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Scott Baldridge ◽  
Adam M. Lowrance ◽  
Ben McCarty
Keyword(s):  
Perfect Matching ◽  
Perfect Matchings ◽  
Trivalent Graph

In this paper, we prove that the $2$-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of $2$-factors that contain the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.  

scholarly journals On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
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.

scholarly journals Face-Width of Pfaffian Braces and Polyhex Graphs on Surfaces

10.37236/3540 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dong Ye ◽  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Face Width ◽  
Graphs On Surfaces ◽  
Positive Genus ◽  
Pfaffian Graph

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 $k\ge 6$.

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

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.

scholarly journals On the König deficiency of zero-reducible graphs

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.

Rainbow Perfect Matchings in Complete Bipartite Graphs: Existence and Counting

2013 ◽  
Vol 22 (5) ◽  
pp. 783-799 ◽  
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.

Enumeration of Architectures With Perfect Matchings

2017 ◽  
Vol 139 (5) ◽  
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.

scholarly journals Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

10.37236/9143 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Marién Abreu ◽  
John Baptist Gauci ◽  
Domenico Labbate ◽  
Giuseppe Mazzuoccolo ◽  
Jean Paul Zerafa
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Perfect Matchings ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Open Problems ◽  
Hamiltonian Property

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

scholarly journals Low Weight Perfect Matchings

10.37236/9994 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Stefan Ehard ◽  
Elena Mohr ◽  
Dieter Rautenbach
Keyword(s):  
Positive Integer ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Low Weight

Answering a question posed by Caro, Hansberg, Lauri, and Zarb, we show that for every positive integer $n$ and every function $\sigma\colon E(K_{4n})\to\{-1,1\}$ with $\sigma\left(E(K_{4n})\right)=0$, there is a perfect matching $M$ in $K_{4n}$with $\sigma(M)=0$. Strengthening the consequence of a result of Caro and Yuster, we show that for every positive integer $n$ and every function $\sigma\colon E(K_{4n})\to\{-1,1\}$ with $\left|\sigma\left(E(K_{4n})\right)\right|<n^2+11n+2,$ there is a perfect matching $M$ in $K_{4n}$ with $|\sigma(M)|\leq 2$. Both these results are best possible.

Fractional Matching Preclusion for (n, k)-Star Graphs

2018 ◽  
Vol 28 (04) ◽  
pp. 1850017 ◽  
Author(s):  
Tianlong Ma ◽  
Yaping Mao ◽  
Eddie Cheng ◽  
Jinling Wang
Keyword(s):  
Perfect Matching ◽  
Perfect Matchings ◽  
Star Graphs ◽  
Minimum Number ◽  
Fractional Perfect Matching

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 introduced the concept of fractional matching preclusion number in 2017. The Fractional Matching Preclusion Number (FMP number) of G 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 G is the minimum number of vertices and/or edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the FMP number and the FSMP number for (n, k)-star graphs. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.

A CHARACTERIZATION OF PM-COMPACT CLAW-FREE CUBIC GRAPHS

2014 ◽  
Vol 06 (02) ◽  
pp. 1450025 ◽  
Author(s):  
XIUMEI WANG ◽  
WEIPING SHANG ◽  
YIXUN LIN ◽  
MARCELO H. CARVALHO
Keyword(s):  
Convex Hull ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Matching Polytopes ◽  
Matching Polytope

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. This paper characterizes claw-free cubic graphs whose 1-skeleton graphs of perfect matching polytopes have diameter 1.

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