scholarly journals A Conjecture of Norine and Thomas for Abelian Cayley Graphs

10.37236/5456 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Fuliang Lu ◽  
Lianzhu Zhang
Keyword(s):  
Polynomial Time ◽  
Perfect Matching ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Perfect Matchings ◽  
Pfaffian Graphs

A graph $\Gamma_1$ is a matching minor of $\Gamma$ if some even subdivision of $\Gamma_1$ is isomorphic to a subgraph $\Gamma_2$ of $\Gamma$, and by deleting the vertices of $\Gamma_2$ from $\Gamma$ the left subgraph has a perfect matching. Motivated by the study of Pfaffian graphs (the numbers of perfect matchings of these graphs can be computed in polynomial time), we characterized Abelian Cayley graphs which do not contain a $K_{3,3}$ matching minor. Furthermore, the Pfaffian property of Cayley graphs on Abelian groups is completely characterized. This result confirms that the conjecture posed by Norine and Thomas in 2008 for Abelian Cayley graphs is true.

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 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.

Some properties of subgroup complementary addition Cayley graphs on abelian groups

2021 ◽  
Author(s):  
Naveen Palanivel ◽  
Chithra A. Velu
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Graph Invariants

In this paper, we introduce subgroup complementary addition Cayley graph [Formula: see text] and compute its graph invariants. Also, we prove that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text] where [Formula: see text].

Invariant random perfect matchings in Cayley graphs

2017 ◽  
Vol 11 (1) ◽  
pp. 211-243 ◽  
Author(s):  
Endre Csóka ◽  
Gabor Lippner
Keyword(s):  
Cayley Graphs ◽  
Perfect Matchings

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.

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