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THE COMPLEXITY OF THOMASON’S ALGORITHM FOR FINDING A SECOND HAMILTONIAN CYCLE

Part of: Graph theory

Published online by Cambridge University Press:  03 May 2018

LIANG ZHONG Open the ORCID record for LIANG ZHONG [Opens in a new window]
LIANG ZHONG*
Affiliation:
Center for Discrete Mathematics, Fuzhou University, Fujian-Fuzhou, China email zhongliangll@126.com
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Abstract

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By Smith’s theorem, if a cubic graph has a Hamiltonian cycle, then it has a second Hamiltonian cycle. Thomason [‘Hamilton cycles and uniquely edge-colourable graphs’, Ann. Discrete Math.3 (1978), 259–268] gave a simple algorithm to find the second cycle. Thomassen [private communication] observed that if there exists a polynomially bounded algorithm for finding a second Hamiltonian cycle in a cubic cyclically 4-edge connected graph $G$, then there exists a polynomially bounded algorithm for finding a second Hamiltonian cycle in any cubic graph $G$. In this paper we present a class of cyclically 4-edge connected cubic bipartite graphs $G_{i}$ with $16(i+1)$ vertices such that Thomason’s algorithm takes $12(2^{i}-1)+3$ steps to find a second Hamiltonian cycle in $G_{i}$.

Keywords

Hamiltonian cyclelollipop method

MSC classification

Primary: 05C45: Eulerian and Hamiltonian graphs
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 18 - 26
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

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