4-connected projective-planar graphs are Hamiltonian-connected

Abstract

We generalize the following two seminal results.(1)Thomassen's result [15] in 1983, which says that every 4-connected planar graph is Hamiltonian-connected (which generalizes the old result of Tutte [16] in 1956, which says that every 4-connected planar graph is Hamiltonian).(2)Thomas and Yu's result [12] in 1994, which says that every 4-connected projective-planar graph is Hamiltonian. Here, Hamiltonian-connected means that for any two vertices u,v, there is a Hamiltonian path between u and v (and hence this generalizes the existence of Hamiltonian cycles).Specifically, we prove the following; This proves a conjecture of Dean [3] in 1990. Our result is best possible in many senses. First, we cannot lower the connectivity 4. Secondly, we cannot generalize our result to a surface with higher genus (that is, there is a 4-connected graph on the torus or on the Klein bottle which is not Hamiltonian-connected).Our proof is constructive in a sense that there is a polynomial time algorithm to find, given two vertices in a 4-connected projective-planar graph, a Hamiltonian path between these two vertices.