scholarly journals An Exploration of the Permanent-Determinant Method

10.37236/1384 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Greg Kuperberg
Keyword(s):  
Bipartite Graph ◽  
Perfect Matchings ◽  
New Techniques ◽  
Enumerative Combinatorics ◽  
Plane Graphs ◽  
Quotient Graph ◽  
Plane Partitions ◽  
Symmetry Classes ◽  
Edge Graph

The permanent-determinant method and its generalization, the Hafnian-Pfaffian method, are methods to enumerate perfect matchings of plane graphs that were discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a bipartite graph on the sphere with $4n$ vertices is invariant under the antipodal map, the number of matchings is the square of the number of matchings of the quotient graph. 2. The number of matchings of the edge graph of a graph with vertices of degree at most 3 is a power of 2. 3. The three Carlitz matrices whose determinants count $a \times b \times c$ plane partitions all have the same cokernel. 4. Two symmetry classes of plane partitions can be enumerated with almost no calculation.

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 $k\ge 6$.

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.

scholarly journals Existence of perfect matchings in a plane bipartite graph

2010 ◽  
Vol 60 (2) ◽  
pp. 489-494
Author(s):  
Zhongyuan Che
Keyword(s):  
Bipartite Graph ◽  
Perfect Matchings

scholarly journals Some hidden relations involving the ten symmetry classes of plane partitions

1994 ◽  
Vol 68 (2) ◽  
pp. 372-409 ◽  
Author(s):  
John R Stembridge
Keyword(s):  
Plane Partitions ◽  
Symmetry Classes

Direct Sum of Distributive Lattices on the Perfect Matchings of a Plane Bipartite Graph

Order ◽  
2010 ◽  
Vol 27 (2) ◽  
pp. 101-113 ◽  
Author(s):  
Heping Zhang
Keyword(s):  
Bipartite Graph ◽  
Perfect Matchings ◽  
Distributive Lattices ◽  
Direct Sum

scholarly journals Plane partitions II: $5\frac{1}{2}$ symmetry classes

2018 ◽  
Author(s):  
Mihai Ciucu ◽  
Christian Krattenthaler
Keyword(s):  
Plane Partitions ◽  
Symmetry Classes

scholarly journals Combinatorial Cluster Expansion Formulas from Triangulated Surfaces

10.37236/8351 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Toshiya Yurikusa
Keyword(s):  
Bipartite Graph ◽  
Cluster Expansion ◽  
Independent Sets ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Expansion Formula ◽  
Triangulated Surfaces ◽  
Quiver With Potential ◽  
Maximal Independent Sets

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.

scholarly journals Signed Lozenge Tilings

10.37236/5579 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
D. Cook II ◽  
Uwe Nagel
Keyword(s):  
Perfect Matchings ◽  
Lattice Paths ◽  
New Method ◽  
Plane Partitions ◽  
Triangular Region ◽  
Lozenge Tilings

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations.

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