Five-Connected Toroidal Graphs Are Hamiltonian

It is well known that not all 3-connected planar graphs are hamiltonian. Whitney [10] proved that every triangulation of the sphere with no separating triangles is hamiltonian. Tutte [9] proved that every 4-connected planar graph has a Hamilton cycle. Extending Tutte's technique, Thomassen [8] proved that every 4-connected planar graph is in fact Hamilton connected. (A small omission in [8] was corrected by Chiba and Nishizeki [3].) With some additional techniques, Thomas and Yu [7] proved that every edge in a 4-connected projective-planar graph is contained in a Hamilton cycle, which establishes a conjecture of Grunbaum [4]. Gru nbaum [4] and Nash-Williams [5] also conjectured that every 4-connected toroidal graph has a Hamilton cycle. While this remains open, Brunet and Richter [2] proved that every 5-connected triangulation of the torus is hamiltonian. Also Altshuler [1] showed that all 6-connected toroidal graphs are Hamiltonian. In this paper we prove the following result.