Hamiltonian cycles in Dirac graphs

Abstract

We prove that for any n-vertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least

$$ exp_2 [2h(G) - n\log e - o(n)], $$

where h(G)=maxΣ e x e log(1/x e ), the maximum over x: E → ℜ+ satisfying Σ eυ x e = 1 for each υV, and log =log2. (A second paper will show that this bound is tight up to the o(n).)

We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1)))n. In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1))n, confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.

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Correspondence to Bill Cuckler.

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Supported by NSF grant DMS0200856.

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Cuckler, B., Kahn, J. Hamiltonian cycles in Dirac graphs. Combinatorica 29, 299–326 (2009). https://doi.org/10.1007/s00493-009-2360-2

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Mathematics Subject Classification (2000)

  • 05A16
  • 05C38
  • 05C45
  • 05D40
  • 05C70