4-connected projective-planar graphs are hamiltonian-connected

Abstract

We generalize the following two seminal results. 1. Thomassen's result [19] in 1983, which says that every 4-connected planar graph is hamiltonian-connected (which generalizes the old result of Tutte [20] in 1956, which says that every 4-connected planar graph is hamiltonian). 2. Thomas and Yu's result [16] 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; Every 4-connected projective planar graph is hamiltonian-connected. 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 (i.e, there is a 4-connected graph on the torus which is not hamiltonian-connected). Our proof is constructive in the sense that there is a polynomial time (in fact, O(n2) time) algorithm to find, given two vertices in a 4-connected projective planar graph, a hamiltonian path between these two vertices.