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

Published online by Cambridge University Press:  03 May 2018

LIANG ZHONG*
Affiliation:
Center for Discrete Mathematics, Fuzhou University, Fujian-Fuzhou, China email [email protected]
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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 $G_{i}$.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

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