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.

Hamiltonian Paths and Cycles

2013 ◽  
pp. 96-105
Author(s):  
Mahtab Hosseininia ◽  
Faraz Dadgostari
Keyword(s):  
Hamiltonian Cycle ◽  
Travelling Salesman Problem ◽  
Sufficient Conditions ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Hamiltonian Graphs ◽  
Hamiltonian Paths ◽  
Graph Problems ◽  
History Of ◽  
Knight’S Tour

In this chapter, the concepts of Hamiltonian paths and Hamiltonian cycles are discussed. In the first section, the history of Hamiltonian graphs is described, and then some concepts such as Hamiltonian paths, Hamiltonian cycles, traceable graphs, and Hamiltonian graphs are defined. Also some most known Hamiltonian graph problems such as travelling salesman problem (TSP), Kirkman’s cell of a bee, Icosian game, and knight’s tour problem are presented. In addition, necessary and (or) sufficient conditions for existence of a Hamiltonian cycle are investigated. Furthermore, in order to solve Hamiltonian cycle problems, some algorithms are introduced in the last section.

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 Perfect matchings in graphs with prescribed local restrictions

2019 ◽  
Vol 63 (4) ◽  
pp. 408-420
Author(s):  
Pavel A. Irzhavski ◽  
Yury L. Orlovich
Keyword(s):  
Regular Graph ◽  
Classical Result ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Perfect Matchings

A graph is called K1,p-restricted (p ≥ 3) if for every vertex of the graph there are at least p - 2 edges between any p neighbours of the vertex. In this article, new sufficient conditions for existence of a perfect matching in K1,p-restricted graphs are established. In particular, J. Petersen’s classical result that every 2-edge connected 3-regular graph contains a perfect matching follows from these conditions.

scholarly journals A Quantitative Approach to Perfect One-Factorizations of Complete Bipartite Graphs

10.37236/4122 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Natacha Astromujoff ◽  
Martin Matamala
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Quantitative Approach ◽  
Hamiltonian Cycles ◽  
Complete Bipartite Graphs

Given a one-factorization $\mathcal{F}$ of the complete bipartite graph $K_{n,n}$, let ${\sf pf}(\mathcal{F})$ denote the number of Hamiltonian cycles obtained by taking pairwise unions of perfect matchings in $\mathcal{F}$. Let ${\sf pf}(n)$ be the maximum of ${\sf pf}(\mathcal{F})$ over all one-factorizations $\mathcal{F}$ of $K_{n,n}$. In this work we prove that ${\sf pf}(n)\geq n^2/4$, for all $n\geq 2$.

Hamiltonian Cycles in Products of Graphs

1975 ◽  
Vol 17 (5) ◽  
pp. 763-765 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Complete Bipartite Graph ◽  
Hamiltonian Path ◽  
Hamiltonian Cycles ◽  
Vertex Set ◽  
Cycle Path

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.

On the Number of Hamiltonian Cycles in Bipartite Graphs

1996 ◽  
Vol 5 (4) ◽  
pp. 437-442 ◽  
Author(s):  
Carsten Thomassen
Keyword(s):  
Hamiltonian Cycle ◽  
Reduction Method ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Hamiltonian Cycles ◽  
Hamiltonian Graph ◽  
Hamiltonian Graphs ◽  
Color Class ◽  
General Reduction

We prove that a bipartite uniquely Hamiltonian graph has a vertex of degree 2 in each color class. As consequences, every bipartite Hamiltonian graph of minimum degree d has at least 21−dd! Hamiltonian cycles, and every bipartite Hamiltonian graph of minimum degree at least 4 and girth g has at least (3/2)g/8 Hamiltonian cycles. We indicate how the existence of more than one Hamiltonian cycle may lead to a general reduction method for Hamiltonian graphs.

Perfect matchings and K1,p-restricted graphs

2020 ◽  
Vol 30 (6) ◽  
pp. 391-408
Author(s):  
Pavel A. Irzhavski ◽  
Yury L. Orlovich
Keyword(s):  
Regular Graph ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Perfect Matchings ◽  
Vertex Degrees

AbstractA graph is called K1,p-restricted (p ≥ 3) if for every vertex of the graph there are at least p − 2 edges between any p of its neighbours. We establish sufficient conditions for the existence of a perfect matching in K1,p-restricted graphs in terms of their connectivity and vertex degrees. These conditions imply, in particular, the classical Petersen’s result: any 2-edge-connected 3-regular graph contains a perfect matching.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

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 Bipartite Consensus of High-Order Edge Dynamics on Coopetition Multiagent Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Yilong Yang ◽  
Zhijian Ji ◽  
Lei Tian ◽  
Huizi Ma ◽  
Qingyuan Qi
Keyword(s):  
Multiagent Systems ◽  
Line Graph ◽  
Sufficient Conditions ◽  
High Order ◽  
Third Order ◽  
Positive Weight ◽  
The Third ◽  
Initial Graph ◽  
Strongly Connected ◽  
Bipartite Consensus

The bipartite consensus of high-order edge dynamics is investigated for coopetition multiagent systems, in which the cooperative and competitive relationships among agents are characterized by positive weight and negative weight, respectively. By mapping the initial graph to a line graph, the distributed control protocol is proposed for the strongly connected, digon sign-symmetric structurally balanced line graph; and then we give sufficient conditions for the third-order multi-gent system to achieve both the bipartite consensus of edge dynamics and the final value of bipartite consensus. By transforming the coefficients of characteristic polynomial from complex domain to real number domain, the sufficient conditions for the bipartite consensus of high-order edge dynamics are also proposed, and the final values of the high-order edge dynamics on multiagent systems are obtained.

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