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

## Abstract

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 $G_{i}$ .

## References

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[1] Cameron, K., ‘Thomason’s algorithm for finding a second hamiltonian circuit through a given edge in a cubic graph is exponential on Krawczyk’s graphs’, Discrete Math. 235 (2001), 6977.
[2] Karp, R. M., ‘Reducibility among combinatorial problems’, in: Complexity of Computer Computations (eds. Miller, R. E., Thatcher, J. W. and Bohlinger, J. D.) (Plenum Press, New York, 1972), 85103.
[3] Krawczyk, A., ‘The complexity of finding a second Hamiltonian cycle in cubic graphs’, J. Comput. Systems Sci. 58 (1999), 641647.
[4] Thomason, A. G., ‘Hamilton cycles and uniquely edge-colourable graphs’, Ann. Discrete Math. 3 (1978), 259268.
[5] Tutte, W. T., ‘On Hamiltonian circuits’, J. Lond. Math. Soc. 21 (1946), 98101.
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# THE COMPLEXITY OF THOMASON’S ALGORITHM FOR FINDING A SECOND HAMILTONIAN CYCLE

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