Hamiltonian embeddings from triangulations

Abstract

A Hamiltonian embedding of Kn is an embedding of Kn in a surface, which may be orientable or non-orientable, in such a way that the boundary of each face is a Hamiltonian cycle. Ellingham and Stephens recently established the existence of such embeddings in non-orientable surfaces for n = 4 and n ⩾ 6. Here we present an entirely new construction which produces Hamiltonian embeddings of Kn from triangulations of Kn when n≡ 0 or 1 (mod 3). We then use this construction to obtain exponential lower bounds for the numbers of nonisomorphic Hamiltonian embeddings of Kn.