A graph G is locally P, abbreviated LP, if for every vertex v in G the open neighbourhood N (v) of v is non-empty and induces a graph with property P. Specifically, a graph G without isolated vertices is locally connected (LC) if N (v) induces a connected graph for each v ∈ V (G), and locally hamiltonian (LH) if N (v) induces a hamiltonian graph for each v ∈ V (G). A graph G is locally locally P (abbreviated L 2 P) if N (v) is non-empty and induces a locally P graph for every v ∈ V (G). This

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... cept is generalized to an arbitrary degree of nesting. For any k ≥ 0 we call a graph locally k-nested-hamiltonian if it is L m C for m = 0, 1, . . . , k and L k H (with L 0 C and L 0 H meaning connected and hamiltonian, respectively). The class of locally k-nested-hamiltonian graphs contains important subclasses. For example, Skupień had already observed in 1963 that the class of connected LH graphs (which is the class of locally 1-nested-hamiltonian graphs) contains all triangulations of closed surfaces. We show that for any k ≥ 1 the class of locally k-nested-hamiltonian graphs contains all simple-clique (k + 2)trees. In 1979 Oberly and Sumner proved that every connected K 1,3 -free graph that is locally connected is hamiltonian. They conjectured that for k ≥ 1, every connected K 1,k+3 -free graph that is locally (k + 1)-connected is hamiltonian. We show that locally k-nested-hamiltonian graphs are locally (k + 1)-connected and consider the weaker conjecture that every K 1,k+3 -free graph that is locally k-nested-hamiltonian is hamiltonian. We show that if our conjecture is true, it would be "best possible" in the sense that for every k ≥ 1 there exist K 1,k+4 -free locally k-nested-hamiltonian graphs that are non-hamiltonian. We also attempt to determine the minimum order of non-hamiltonian locally k-nested-hamiltonian graphs and investigate the complexity of the Hamilton Cycle Problem for locally k-nested-hamiltonian graphs with restricted maximum degree. J.P. de Wet and M. Frick