# Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle

In this paper we consider the following total functional problem: Given a cubic Hamiltonian graph G and a Hamiltonian cycle C_0 of G, how can we compute a second Hamiltonian cycle C_1 ≠ C_0 of G? Cedric Smith proved in 1946, using a non-constructive parity argument, that such a second Hamiltonian cycle always exists. Our main result is an algorithm which computes the second Hamiltonian cycle in time O(n · 2^(0.3-ε)n) time, for some positive constant ε>0, and in polynomial space, thus improving the state of the art running time for solving this problem. Our algorithm is based on a fundamental structural property of Thomason's lollipop algorithm, which we prove here for the first time. In the direction of approximating the length of a second cycle in a Hamiltonian graph G with a given Hamiltonian cycle C_0 (where we may not have guarantees on the existence of a second Hamiltonian cycle), we provide a linear-time algorithm computing a second cycle with length at least n - 4α (√(n)+2α)+8, where α = Δ-2/δ-2 and δ,Δ are the minimum and the maximum degree of the graph, respectively. This approximation result also improves the state of the art.

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