scholarly journals On packing Hamilton cycles in ε-regular graphs

2005 ◽  
Vol 94 (1) ◽  
pp. 159-172 ◽  
Author(s):  
Alan Frieze ◽  
Michael Krivelevich
Keyword(s):  
Regular Graphs ◽  
Hamilton Cycles

Hamilton cycles containing randomly selected edges in random regular graphs

2001 ◽  
Vol 19 (2) ◽  
pp. 128-147 ◽  
Author(s):  
R.W. Robinson ◽  
N.C. Wormald
Keyword(s):  
Regular Graphs ◽  
Hamilton Cycles

scholarly journals Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

2012 ◽  
Vol 22 (3) ◽  
pp. 394-416 ◽  
Author(s):  
DANIELA KÜHN ◽  
JOHN LAPINSKAS ◽  
DERYK OSTHUS
Keyword(s):  
General Result ◽  
Minimum Degree ◽  
Regular Graphs ◽  
Hamilton Cycles ◽  
Edge Disjoint ◽  
Large Graph ◽  
Spanning Subgraph ◽  
Hamilton Decomposition

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ=(1/2+α)n. For any constant α>0, we give an optimal answer in the following sense: let regeven(n,δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals regeven(n,δ)/2. The value of regeven(n,δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

scholarly journals Generating and Counting Hamilton Cycles in Random Regular Graphs

1996 ◽  
Vol 21 (1) ◽  
pp. 176-198 ◽  
Author(s):  
Alan Frieze ◽  
Mark Jerrum ◽  
Michael Molloy ◽  
Robert Robinson ◽  
Nicholas Wormald
Keyword(s):  
Regular Graphs ◽  
Hamilton Cycles

scholarly journals Rainbow Hamilton cycles in random regular graphs

2006 ◽  
Vol 30 (1-2) ◽  
pp. 35-49 ◽  
Author(s):  
Svante Janson ◽  
Nicholas Wormald
Keyword(s):  
Regular Graphs ◽  
Hamilton Cycles

scholarly journals Solution to a problem of Bollobás and Häggkvist on Hamilton cycles in regular graphs

2016 ◽  
Vol 121 ◽  
pp. 85-145 ◽  
Author(s):  
Daniela Kühn ◽  
Allan Lo ◽  
Deryk Osthus ◽  
Katherine Staden
Keyword(s):  
Regular Graphs ◽  
Hamilton Cycles

scholarly journals Enumerating all Hamilton Cycles and Bounding the Number of Hamilton Cycles in 3-Regular Graphs

10.37236/619 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Heidi Gebauer
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Hamilton Cycle ◽  
Regular Graphs ◽  
Hamilton Cycles ◽  
Vertex Graph

We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular $n$-vertex graph in time $O(1.276^{n})$, improving on Eppstein's previous bound. The resulting new upper bound of $O(1.276^{n})$ for the maximum number of Hamilton cycles in 3-regular $n$-vertex graphs gets close to the best known lower bound of $\Omega(1.259^{n})$. Our method differs from Eppstein's in that he considers in each step a new graph and modifies it, while we fix (at the very beginning) one Hamilton cycle $C$ and then proceed around $C$, successively producing partial Hamilton cycles.

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