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We investigate the minimum number t 0 ( G ) of faces in a Hamiltonian triangulation G so that any Hamiltonian cycle C of G has at least t 0 ( G ) faces that do not contain an edge of C . We prove upper and lower bounds on the maximum of these numbers for all triangulations with a fixed number of facial triangles. Such triangles play an important role when Hamiltonian cycles in triangulations with 3 -cuts are constructed from smaller Hamiltonian cycles of 4 -connected subgraphs. We also present results linking the number of these triangles to the length of 3 -walks in a class of triangulation and to the domination number.

Types of triangle in Hamiltonian triangulations and an application to domination and k-walks