scholarly journals Covering a cubic graph by 5 perfect matchings

2017 ◽  
Vol 340 (8) ◽  
pp. 1889-1896
Author(s):  
Wuyang Sun
Keyword(s):  
Perfect Matchings ◽  
Cubic Graph

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.

scholarly journals Perfect Matching Covers of Cubic Graphs of Oddness 2

10.37236/7175 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Wuyang Sun ◽  
Fan Wang
Keyword(s):  
Perfect Matching ◽  
Perfect Matchings ◽  
Cubic Graphs ◽  
Cubic Graph

A perfect matching cover of a graph $G$ is a set of perfect matchings of $G$ such that each edge of $G$ is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph has a perfect matching cover of order at most 5. The Berge Conjecture is largely open and it is even unknown whether a constant integer $c$ does exist such that every bridgeless cubic graph has a perfect matching cover of order at most $c$. In this paper, we show that a bridgeless cubic graph $G$ has a perfect matching cover of order at most 11 if $G$ has a 2-factor in which the number of odd circuits is 2.

scholarly journals On Snarks that are far from being 3-Edge Colorable

10.37236/3430 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jonas Hägglund
Keyword(s):  
Double Cover ◽  
Interesting Property ◽  
Perfect Matchings ◽  
Petersen Graph ◽  
Cubic Graph

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.

scholarly journals Covering a cubic graph with perfect matchings

2013 ◽  
Vol 313 (20) ◽  
pp. 2292-2296
Author(s):  
G. Mazzuoccolo
Keyword(s):  
Perfect Matchings ◽  
Cubic Graph

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 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.

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