For a given graph property $$\mathcal {P}$$ P , we say a graph G is locally $$\mathcal {P}$$ P if for each $$v \in V(G)$$ v ∈ V ( G ) , the subgraph induced by the open neighbourhood of v has property $$\mathcal P$$ P . A closed locally $$\mathcal {P}$$ P graph is defined analogously in terms of closed neighbourhoods. It is known that connected locally hamiltonian graphs are not necessarily hamiltonian. Saito (in Computational Geometry and Graph Theory, Lecture Notes in Computer Science, vol. 4535, pp. 191–200. Springer, Berlin, 2008) conjectured that if G is a graph of order at least 3 such that for every vertex v in G the subgraph induced by the closed neighbourhood N[v] of v satisfies the Chvátal–Erdős condition for hamiltonicity, then G is hamiltonian. Oberly and Sumner (in J Graph Theory 3:351–356, 1979) conjectured that if G is a connected, locally k-connected $$K_{1,k+2}$$ K 1 , k + 2 -free graph of order at at least 3, then G is hamiltonian. We prove a result that lends support to both these conjectures. We also provide a framework for investigating these and other related conjectures.