Hamiltonian cycles in graphs of triangular grid

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. It is shown that all connected, locally connected triangular grid graphs on, at least, three vertices are fully cyclic extendable (with the only one exception). This result generalizes J. Reay and T. Zamfirescu's theorem on the hamiltonicity of 2-connected, linearly convex triangular grid graphs. Recognition of the hamiltonicity for triangular grid graphs in the general case is established to be NP-complete, but the hamiltonian cycle in the connected, locally connected triangular grid graph can be found in polynomial time.