scholarly journals Enumeration of Perfect Matchings of a Type of Quadratic Lattice on the Torus

10.37236/308 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Fuliang Lu ◽  
Lianzhu Zhang ◽  
Fenggen Lin
Keyword(s):  
Cartesian Product ◽  
Perfect Matchings ◽  
Quadratic Lattice ◽  
Explicit Expressions

A quadrilateral cylinder of length $m$ and breadth $n$ is the Cartesian product of a $m$-cycle(with $m$ vertices) and a $n$-path(with $n$ vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as $x_1,x_2,\cdots,x_m$ and $y_1,y_2,\cdots ,y_m$, respectively, where $x_i$ corresponds to $y_i(i=1,2,\dots, m)$. We denote by $Q_{m,n,r}$, the graph obtained from quadrilateral cylinder of length $m$ and breadth $n$ by adding edges $x_iy_{i+r}$ ($r$ is a integer, $0\leq r < m$ and $i+r$ is modulo $m$). Kasteleyn had derived explicit expressions of the number of perfect matchings for $Q_{m,n,0}$ [P.W. Kasteleyn, The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice, Physica 27(1961), 1209–1225]. In this paper, we generalize the result of Kasteleyn, and obtain expressions of the number of perfect matchings for $Q_{m,n,r}$ by enumerating Pfaffians.

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 $k\ge 6$.

STRONG MATCHING PRECLUSION FOR THE ALTERNATING GROUP GRAPHS AND SPLIT-STARS

2011 ◽  
Vol 12 (04) ◽  
pp. 277-298 ◽  
Author(s):  
PHILIP BONNEVILLE ◽  
EDDIE CHENG ◽  
JOSEPH RENZI
Keyword(s):  
General Result ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Alternating Group ◽  
Optimal Solutions ◽  
Minimum Number

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem and has recently been introduced by Park and Ihm.15 In this paper, we examine properties of strong matching preclusion for alternating group graphs, by finding their strong matching preclusion numbers and categorizing all optimal solutions. More importantly, we prove a general result on taking a Cartesian product of a graph with K2 (an edge) to obtain the corresponding results for split-stars.

Complete forcing numbers of rectangular polynominoes

2021 ◽  
Vol 1 (1) ◽  
pp. 87-96
Author(s):  
Hong Chang ◽  
Yong-De Feng ◽  
Hong Bian ◽  
Shou-Jun Xu
Keyword(s):  
Perfect Matching ◽  
Cartesian Product ◽  
Perfect Matchings ◽  
Forcing Number ◽  
Explicit Formulae ◽  
Hexagonal Systems ◽  
Forcing Set ◽  
Complete Forcing Number

Let G be a graph with edge set E(G) that admits a perfect matching M. A forcing set of M is a subset of M contained in no other perfect matchings of G. A complete forcing set of G, recently introduced by Xu et al. [Complete forcing numbers of catacondensed hexagonal systems, J. Combin. Optim. 29(4) (2015) 803-814], is a subset of E(G) on which the restriction of any perfect matching M is a forcing set of M. The minimum possible cardinality of complete forcing sets of G is the complete forcing number of G. In this article, we discuss the complete forcing number of rectangular polyominoes (or grids), i.e., the Cartesian product of two paths of various lengths, and show explicit formulae for the complete forcing numbers of rectangular polyominoes in terms of the lengths.

scholarly journals Pfaffian Orientation and Enumeration of Perfect Matchings for some Cartesian Products of Graphs

10.37236/141 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Feng-Gen Lin ◽  
Lian-Zhu Zhang
Keyword(s):  
Cartesian Product ◽  
Perfect Matchings ◽  
Upper And Lower Bounds ◽  
Cartesian Products ◽  
Pfaffian Orientation ◽  
Enumeration Problem ◽  
General Graphs ◽  
Skew Adjacency Matrix ◽  
Product Of Graphs ◽  
Equivalent Conditions

The importance of Pfaffian orientations stems from the fact that if a graph $G$ is Pfaffian, then the number of perfect matchings of $G$ (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian. Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP. It is well known that the enumeration problem of perfect matchings for general graphs is NP-hard. L. Lovász pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly. For a simple graph $G$ and a cycle $C_n$ with $n$ vertices (or a path $P_n$ with $n$ vertices), we define $C_n$ (or $P_n)\times G$ as the Cartesian product of graphs $C_n$ (or $P_n$) and $G$. In the present paper, we construct Pfaffian orientations of graphs $C_4\times G$, $P_4\times G$ and $P_3\times G$, where $G$ is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of $\overrightarrow{G}$ to enumerate their perfect matchings by Pfaffian approach, where $\overrightarrow{G}$ is an arbitrary orientation of $G$.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

CARTESIAN PRODUCT ON FUZZY IDEALS OF A TERNARY Γ-SEMIGROUP

2020 ◽  
Vol 9 (3) ◽  
pp. 1189-1195 ◽  
Author(s):  
Y. Bhargavi ◽  
T. Eswarlal ◽  
S. Ragamayi
Keyword(s):  
Cartesian Product

scholarly journals THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

2020 ◽  
pp. 1-10
Author(s):  
JING TIAN ◽  
KEXIANG XU ◽  
SANDI KLAVŽAR
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
General Position ◽  
Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

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