Hamiltonian cycles in 4-connected plane triangulations with few 4-separators

Hakimi, Schmeichel and Thomassen showed in 1979 that every 4-connected triangu-lation on n vertices has at least n/log(2) n hamiltonian cycles, and conjectured that the sharp lower bound is 2(n - 2)(n - 4). Recently, Brinkmann, Souffriau and Van Cleemput gave an improved lower bound 12/5 (n - 2). In this paper we show that every 4-connected triangulation with O(log n) 4-separators has Omega(n(2)/log(2) n) hamiltonian cycles. (c) 2020 Elsevier B.V. All rights reserved.