The Hamiltonian Cycle Problem is Linear-Time Solvable for 4-Connected Planar Graphs

A Hamiltonian cycle (path) of a graph G is a simple cycle (path) which contains all the vertices of G. The Hamiltonian cycle problem asks whether a given graph contains a Hamiltonian cycle. It is NP-complete even for 3-connected planar graphs [3, 61. However, the problem becomes polynomial-time solvable for Cconnected planar graphs: Tutte proved that such a graph necessarily contains a Hamiltonian cycle [9, lo]; and, moreover, Gouyou-Beauchamps [4], based on Tutte’s proof, gave an O(n3) algorithm which actually finds a Hamiltonian cycle in such a graph. Throughout the paper n denotes the number of vertices in a graph. In this paper we give a linear algorithm for finding a Hamiltonian cycle in Cconnected planar graphs. This linear algorithm improves GouyouBeauchamps’ O(n3) and Asano, Kikuchi, and Saito’s linear algorithms [l]; the last works only for 4-comected maximal planar graphs.