4-connected Projective-planar Graphs Are Hamiltonian-connected

We generalize the following two seminal results. 1. Thomassen’s result [19] in 1983, which says thatevery 4-connected planar graph is hamiltonian-connected (which generalizes the old result of Tutte[20] in 1956, which says that every 4-connectedplanar graph is hamiltonian). 2. Thomas and Yu’s result [16] in 1994, which saysthat every 4-connected projective planar graph ishamiltonian. Here, hamiltonian-connected means that for any twovertices u, v, there is a hamiltonian path between uand v (and hence this generalizes the existence ofhamiltonian cycles).Specifically, we prove the following; Every 4-connected projective planar graph ishamiltonian-connected. This proves a conjecture of Dean [3] in 1990. Ourresult is best possible in many senses. First, we cannotlower the connectivity 4. Secondly, we cannot generalizeour 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 sensethat there is a polynomial time (in fact, O(n) time)algorithm to find, given two vertices in a 4-connectedprojective planar graph, a hamiltonian path betweenthese two vertices.