Empirical Study of Phase Transition of Hamiltonian Cycle Problem in Random Graphs with Degrees Greater Than One

2016 ◽  
pp. 194-204
Author(s):  
Wei Peng ◽  
Dongxia Wang ◽  
Xinwen Jiang
Keyword(s):  
Phase Transition ◽  
Empirical Study ◽  
Random Graphs ◽  
Hamiltonian Cycle ◽  
Hamiltonian Cycle Problem

scholarly journals A random hierarchical lattice: the series-parallel graph and its properties

2004 ◽  
Vol 36 (03) ◽  
pp. 824-838 ◽  
Author(s):  
B. M. Hambly ◽  
Jonathan Jordan
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Effective Resistance ◽  
First Passage Percolation ◽  
Hierarchical Lattice ◽  
First Passage ◽  
In Series ◽  
Parallel Graph

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

scholarly journals The Size of the Giant Joint Component in a Binomial Random Double Graph

10.37236/8846 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Mark Jerrum ◽  
Tamás Makai
Keyword(s):  
Phase Transition ◽  
Critical Point ◽  
Random Graphs ◽  
Spanning Tree ◽  
Branching Process ◽  
Graph Model ◽  
First Order ◽  
Blue Edge ◽  
Critical Pairs

We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices.  A joint component is a maximal set of vertices that supports both a red and a blue spanning tree.  We show that there are critical pairs of red and blue edge densities at which a giant joint component appears.  In contrast to the standard binomial graph model, the phase transition is first order:  the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point.  We connect this phenomenon to the properties of a certain bicoloured branching process. 

scholarly journals The Gn,m Phase Transition is Not Hard for the Hamiltonian Cycle Problem

1998 ◽  
Vol 9 ◽  
pp. 219-245 ◽  
Author(s):  
B. Vandegriend ◽  
J. Culberson
Keyword(s):  
Phase Transition ◽  
High Frequency ◽  
Maximum Degree ◽  
Hamiltonian Cycle ◽  
Degree Sequence ◽  
Average Degree ◽  
Regular Graphs ◽  
M Phase ◽  
Backtrack Algorithm ◽  
Pruning Techniques

Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian Cycle instances with the Gn,m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the Gn,m phase transition which asymptotically produces a high frequency of hard instances.

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