Finding large cycles in Hamiltonian graphs

  • Authors:

  • Tomás Feder;Rajeev Motwani

  • Affiliations:

  • 268 Waverley Street, Palo Alto, CA 94301, United States;Department of Computer Science, Stanford University, Stanford, CA 94305, United States

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

We show how to find in Hamiltonian graphs a cycle of length n^@W^(^1^/^l^o^g^l^o^g^n^)=exp(@W(logn/loglogn)). This is a consequence of a more general result in which we show that if G has a maximum degree d and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in O(n^3) time a cycle in G of length k^@W^(^1^/^l^o^g^d^). From this we infer that if G has a cycle of length k, then one can find in O(n^3) time a cycle of length k^@W^(^1^/^(^l^o^g^(^n^/^k^)^+^l^o^g^l^o^g^n^)^), which implies the result for Hamiltonian graphs. Our results improve, for some values of k and d, a recent result of Gabow (2004) [11] showing that if G has a cycle of length k, then one can find in polynomial time a cycle in G of length exp(@W(logk/loglogk)). We finally show that if G has fixed Euler genus g and has a cycle with k vertices (or a 3-cyclable minor H with k vertices), then we can find in polynomial time a cycle in G of length f(g)k^@W^(^1^), running in time O(n^2) for planar graphs.