Traceability of locally hamiltonian and locally traceable graphs

If $mathcal{P}$ is a given graph property, we say that a click here graph $G$ is locally $mathcal{P}$ if $langle N(v) angle$ has property $mathcal{P}$ for every $v in V(G)$ where $langle N(v) angle$ is the induced graph on the open neighbourhood of the vertex $v$.Pareek and Skupien (C.M.Pareek and Z.

Skupien , On the smallest non-Hamiltonian locally Hamiltonian graph, J.Univ.Kuwait (Sci.), 10:9 - 17, 1983) posed the following two questions.

Question 1 Is 9 the smallest order of a connected nontraceable locally traceable graph? Question 2 Is 14 click here the smallest order of a connected nontraceable locally hamiltonian graph? We answer the second question in the affirmative, but show that the correct number for the first question is 10.We develop a technique to construct connected locally hamiltonian and locally traceable graphs that are not traceable.We use this technique to construct such graphs with various prescribed properties.