Sufficient and Necessary Conditions for an Edge in the Optimal Hamiltonian Cycle Based on Frequency Quadrilaterals

  • Yong WangEmail author


The symmetric traveling salesman problem is studied according to frequency graphs computed with frequency quadrilaterals. Here, we provide the sufficient and necessary conditions for an optimal Hamiltonian cycle edge based on frequency quadrilaterals. If the probability that an edge has frequency 5 in a frequency quadrilateral is 1, it belongs to the optimal Hamiltonian cycle. For an optimal Hamiltonian cycle edge, the probability that it has frequency 5 in a frequency quadrilateral tends to 1 as the scale of traveling salesman problem is sufficiently large.


Traveling salesman problem Optimal Hamiltonian cycle Frequency quadrilateral Sufficient condition Necessary condition 

Mathematics Subject Classification

65K10 68R10 



The authors acknowledge the anonymous referees for their helpful comments to improve the presentation of the paper. We acknowledge Reinelt, G., et al. who provided the TSP data to the TSPLIB, and Cook, W. and Mittelmann, H. who provided the online Concorde. The authors acknowledge the funds supported by the Fundamental Research Funds for the Central Universities (Nos. 2018MS039 and 2018ZD09) and National Natural Science Foundation of China (No. 51205129).


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.North China Electric Power UniversityBeijingChina

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