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Finding large cycles in Hamiltonian graphs

Tomás Feder, Rajeev Motwani
2010 Discrete Applied Mathematics
We show how to find in Hamiltonian graphs a cycle of length n Ω(1/ log log n) .  ...  From this we infer that if G has a cycle of length k, then one can find in O(n 3 ) time a cycle of length k Ω(1/(log(n/k)+log log n)) , which implies the result for Hamiltonian graphs.  ...  Long Cycles in Hamiltonian Graphs We infer from Theorem 5 that one can find long cycles in Hamiltonian graphs. Theorem 6 Let G be an n-vertex graph that has a Hamiltonian cycle.  ...

Finding Uniquely Hamiltonian Graphs of Minimum Degree Three with Small Crossing Numbers [chapter]

Benedikt Klocker, Herbert Fleischner, Günther R. Raidl
2016 Lecture Notes in Computer Science
In this work we try to find uniquely hamiltonian graphs with minimum degree three and a small crossing number by minimizing the number of crossings in an embedding and the number of degree-two vertices  ...  Although our implementation could not find a uniquely hamiltonian planar graph with minimum degree three disproving Bondy and Jackson's conjecture, we were able to find uniquely hamiltonian graphs of minimum  ...  This means that whenever LKH did not find a second hamiltonian cycle the graph was in fact uniquely hamiltonian.  ...

Use of Hamiltonian Cycles in Cryptograph [article]

Hsieh WenBin, Leu Jenq-Shiou
2012 arXiv   pre-print
Ejov et al.[16-18] find out 76,800 Hamiltonian cycles in Horton94, but there are 3 94 possible sub-graphs.  ...  The cycle v 1 v 2 v 5 v 6 v 4 v 3 v 1 forms a Hamiltonian cycle. If one more vertex is added like a graph in Fig.2 .  ...

Deterministic "Snakes and Ladders" Heuristic for the Hamiltonian Cycle Problem [article]

Pouya Baniasadi, Vladimir Ejov, Jerzy A Filar, Michael Haythorpe and Serguei Rossomakhine
2019 arXiv   pre-print
We present a polynomial complexity, deterministic, heuristic for solving the Hamiltonian Cycle Problem (HCP) in an undirected graph of order n.  ...  Although finding a Hamiltonian cycle is not theoretically guaranteed, we have observed that the heuristic is successful even in cases where such cycles are extremely rare, and it also performs very well  ...  The research presented in this manuscript was supported by the ARC Discovery Grant DP120100532.  ...

A new algorithm to find fuzzy Hamilton cycle in a fuzzy network using adjacency matrix and minimum vertex degree

A. Nagoor Gani, S. R. Latha
2016 SpringerPlus
A graph containing a Hamiltonian cycle is called a Hamiltonian graph. There have been several researches to find the number of Hamiltonian cycles of a Hamilton graph.  ...  In this paper, a new algorithm is proposed to find fuzzy Hamiltonian cycle using adjacency matrix and the degree of the vertices of a fuzzy graph.  ...  Algorithm to find fuzzy Hamiltonian cycle in a fuzzy graph G In this section, two algorithms are proposed to find a Fuzzy Hamiltonian cycle in a fuzzy graph.  ...

Heuristics for sparse general travelling salesman problems

Les R. Foulds, Derek R. O'Connor
1992 The Australasian Journal of Combinatorics
The well-known travelling salesman problem can be expressed in graphtheoretic terms as follows : find a minimum-weight Hamiltonian cycle in a connected edge-weighted graph G(N, A). vVe concentrate here  ...  Sparse graphs may not have Hamiltonian cycles and hence there may be no solution to the problem as stated. vVhether this is the case or not, the requirement that the cycle is Hamiltonian is often unnecessarily  ...  A Hamiltonian Cycle of G is a simple cycle in G in \\'hich every node of G appears exactly once. If a graph has a Hamiltonian cycle it is called a Hamiltonian graph.  ...

Evolutionary Hamiltonian Graph Theory [article]

Zh. G. Nikoghosyan
2012 arXiv   pre-print
We present an alternative domain concerning mathematics to investigate universal evolution mechanisms by focusing on large cycles theory (LCT) - a simplified version of well-known hamiltonian graph theory  ...  LCT joins together a number of NP-complete cycle problems in graph theory.  ...  Other types of large cycles were introduced for different situations when the graph contains no Hamilton cycles or it is difficult to find it.  ...

HybridHAM: A Novel Hybrid Heuristic for Finding Hamiltonian Cycle

K. R. Seeja
2018 Journal of Optimization
The proposed approach could find Hamiltonian cycles from a set of hard graphs collected from the literature, all the Hamiltonian instances (1000 to 5000 vertices) given in TSPLIB, and some instances of  ...  Third phase converts the Hamiltonian path into a Hamiltonian cycle by using rotational transformation.  ...  Acknowledgments The research presented in this manuscript is supported by University Grants Commission Major Project Grant F.No.42-136/2013(SR).  ...

Finding long cycles in graphs

Enzo Marinari, Guilhem Semerjian, Valery Van Kerrebroeck
2007 Physical Review E
Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three.  ...  We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task.  ...  Acknowledgments This work was supported by EVERGROW, integrated project No. 1935 in the complex systems initiative of the Future and Emerging Technologies directorate of the IST Priority, EU Sixth Framework  ...

A new benchmark set for Traveling salesman problem and Hamiltonian cycle problem [article]

2018 arXiv   pre-print
These instances are based on difficult instances of Hamiltonian cycle problem (HCP).  ...  We also include the HCP heuristic SLH in the benchmarking exercise.  ...  Step 2: Create random Hamiltonian graph G with a known Hamiltonian Cycle HC i . Step 3: Solve the graph G using the chosen solver and find a Hamiltonian cycle HC r .  ...

Sufficient Conditions for Hamiltonicity in Communication Networks through Graph Modeling: New Research Challenges

Ajeet Singh, Vikas Tiwari
2017 International Journal of Computer Applications
A network graph containing a Hamiltonian Cycle in it, is said to be Hamiltonian. The problem of finding whether a graph G is Hamiltonian is proved to be NP-Complete.  ...  A Hamiltonian Path is a spanning trail in a network graph i.e. a path through every network node and a spanning cycle in a network graph is a Hamiltonian cycle.  ...  A Hamiltonian cycle is a spanning cycle in the graph and a graph containing a Hamiltonian Cycle is said to be Hamiltonian.  ...

AUTOMATICALLY FINDING THE HAMILTONIAN PATH IN A DIRECTED GRAPH WITHOUT CYCLES BY USING THE YU CHEN ALGORITHM

2017 Scientific Bulletin of Naval Academy
"Yu Chen's algorithm" [1] , [3] , [5] solves the problem of finding the Hamiltonian path in a directed graph without cycles.  ...  The algorithm finds the adjacency matrix in a directed, finite graph, with vertices. If the graph has no cycles, the algorithm computes the powers of reaching the vertices.  ...  INTRODUCTION The problem of finding the Hamiltonian paths in a directed graph and without cycles has been solved by Yu Chen's algorithm.  ...

Backtracking (the) Algorithms on the Hamiltonian Cycle Problem [article]

Joeri Sleegers, Daan van den Berg
2021 arXiv   pre-print
Even though the Hamiltonian cycle problem is NP-complete, many of its problem instances aren't.  ...  Recently however, targeted search efforts have identified very hard Hamiltonian cycle problem instances very far away from the Komlós-Szemerédi bound.  ...  Martello's algorithm was designed with the purpose of finding all Hamiltonian cycles in directed graphs.  ...

Page 3544 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews
Christofides, Graph theory, Academic Press, New York, 1975; MR 55 #2623] for finding Hamilton cycles does not work very well for non-Hamiltonian graphs.  ...  Beineke (Fort Wayne, IN) 93g:05092 05C45 05C15 05C38 Chetwynd, A. G. (4-LANC); Hilton, A. J. W. (4-RDNG) Alternating Hamiltonian cycles in two-colored complete bipartite graphs. J.  ...

On Hamiltonian Berge cycles in 3-uniform hypergraphs [article]

Linyuan Lu, Zhiyu Wang
2020 arXiv   pre-print
In this note, we show that every covering [3]-uniform hypergraph on n≥ 6 vertices contains a Berge cycle C_s for any 3≤ s≤ n.  ...  A hypergraph H is called covering if every vertex pair is contained in some hyperedge in H.  ...  The problem of finding Hamiltonian Berge cycles in a hypergraph is closely related to the problem of finding rainbow Hamiltonian cycles in an edge-colored complete graph K n .  ...
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