Exact and Approximate Algorithms for Computing a Second Hamiltonian Cycle

by   Argyrios Deligkas, et al.

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.


page 1

page 2

page 3

page 4


Find Subtrees of Specified Weight and Cycles of Specified Length in Linear Time

We introduce a variant of DFS which finds subtrees of specified weight i...

An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs

We present a polynomial space Monte Carlo algorithm that given a directe...

A Method to Compute the Sparse Graphs for Traveling Salesman Problem Based on Frequency Quadrilaterals

In this paper, an iterative algorithm is designed to compute the sparse ...

A fast algorithm on average for solving the Hamilton Cycle problem

We present CertifyHAM, an algorithm which takes as input a graph G and e...

Approximating Long Cycle Above Dirac's Guarantee

Parameterization above (or below) a guarantee is a successful concept in...

FHCP Challenge Set: The First Set of Structurally Difficult Instances of the Hamiltonian Cycle Problem

The FHCP Challenge Set, comprising of 1001 instances of Hamiltonian cycl...

Fast algorithms for solving the Hamilton Cycle problem with high probability

We study the Hamilton cycle problem with input a random graph G=G(n,p) i...

Please sign up or login with your details

Forgot password? Click here to reset