Hamiltonicity of locally hamiltonian and locally traceable graphs

If P is a given graph property, we say that a graph G is locallyP if 〈N(v)〉 has property P for every v∈V(G) where 〈N(v)〉 is the induced graph on the open neighbourhood of the vertex v. We consider the complexity of the Hamilton Cycle Problem for locally traceable and locally hamiltonian graphs with small maximum degree. The problem is fully solved for locally traceable graphs with maximum degree 5 and also for locally hamiltonian graphs with maximum degree 6 (van Aardt et al., 2016). We show that the Hamilton Cycle Problem is NP-complete for locally traceable graphs with maximum degree 6 and for locally hamiltonian graphs with maximum degree 10. We also show that there exist regular connected nonhamiltonian locally hamiltonian graphs with connectivity 3, thus answering two questions posed by Pareek and Skupień (1983).