Subhamiltonian toroidal graphs

The problem of determining whether a graph contains a Hamiltonian cycle is difficult but has been well-studied. A related question asks when is a graph, embeddable on a surface S, a subgraph of a Hamiltonian graph which is also embeddable on S? In particular, if a graph has genus g, is it a subgraph of a Hamiltonian graph of genus g? We answer this question for all complete graphs and complete m-partite graphs of genus 0 and 1.