scholarly journals A New Lower Bound on the Number of Perfect Matchings in Cubic Graphs

2009 ◽  
Vol 23 (3) ◽  
pp. 1465-1483 ◽  
Author(s):  
Daniel Král' ◽  
Jean-Sébastien Sereni ◽  
Michael Stiebitz
Keyword(s):  
Lower Bound ◽  
Perfect Matchings ◽  
Cubic Graphs

Lower Bounds For Induced Forests in Cubic Graphs

1987 ◽  
Vol 30 (2) ◽  
pp. 193-199 ◽  
Author(s):  
J. A. Bondy ◽  
Glenn Hopkins ◽  
William Staton
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cubic Graphs ◽  
Cubic Graph

AbstractIf G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

scholarly journals The Fan–Raspaud conjecture: A randomized algorithmic approach and application to the pair assignment problem in cubic networks

2012 ◽  
Vol 22 (3) ◽  
pp. 765-778
Author(s):  
Piotr Formanowicz ◽  
Krzysztof Tanaś
Keyword(s):  
Graph Theory ◽  
Computer Networks ◽  
Assignment Problem ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Algorithmic Approach ◽  
Edge Colorings ◽  
Mathematical Properties

Abstract It was conjectured by Fan and Raspaud (1994) that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan–Raspaud colorings of a given cubic graph and, analyzing the computer results, we try to find and describe the Fan–Raspaud colorings for some selected classes of cubic graphs. The presented algorithms can then be applied to the pair assignment problem in cubic computer networks. Another possible application of the algorithms is that of being a tool for mathematicians working in the field of cubic graph theory, for discovering edge colorings with certain mathematical properties and formulating new conjectures related to the Fan–Raspaud conjecture.

PERFECT MATCHINGS AND GENUS OF SOME CARTESIAN PRODUCTS OF GRAPHS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250026
Author(s):  
FENGGEN LIN ◽  
LIANZHU ZHANG ◽  
FULIANG LU
Keyword(s):  
Lower Bound ◽  
Perfect Matchings ◽  
Cartesian Products ◽  
Elementary Graph ◽  
Cartesian Products Of Graphs

Lovász and Plummer conjectured that: for k ≥ 3 there exist constants c1(k) > 1 and c2(k) > 0 such that every k-regular elementary graph on 2n vertices, without forbidden edges, contains at least c2(k)⋅c1(k)nperfect matchings. Furthermore c1(k) → ∞ as k → ∞. In this paper, for some Cartesian products of graphs, we obtain a lower bound of their number of perfect matchings which is similar to that of Lovász and Plummer's conjecture. Furthermore, we compute the genus of some Cartesian products of graphs.

scholarly journals Packing Plane Perfect Matchings into a Point Set

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Ahmad Biniaz ◽  
Prosenjit Bose ◽  
Anil Maheshwari ◽  
Michiel Smid
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Perfect Matchings ◽  
Point Sets ◽  
Exact Answer ◽  
International Audience ◽  
Point Set

International audience Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of &#x230A;log<sub>2</sub>$n$&#x230B;$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.

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.

Unions of perfect matchings in cubic graphs

2005 ◽  
Vol 22 ◽  
pp. 341-345 ◽  
Author(s):  
Tomáš Kaiser ◽  
Daniel Král' ◽  
Serguei Norine
Keyword(s):  
Perfect Matchings ◽  
Cubic Graphs

scholarly journals Counting matchings via capacity-preserving operators

2021 ◽  
pp. 1-26
Author(s):  
Leonid Gurvits ◽  
Jonathan Leake
Keyword(s):  
Differential Operator ◽  
Lower Bound ◽  
Recent Result ◽  
Polynomial Time ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Polynomial Time Algorithms ◽  
Real Stability

Abstract The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

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