Dominating plane triangulations

In 1996, Tarjan and Matheson proved that if G is a plane triangulated disc with n vertices, γ ( G ) ź n / 3 , where γ ( G ) denotes the domination number of G , i.e.źthe cardinality of the smallest set of vertices S such that every vertex of G is either in S or adjacent to a vertex in S . Furthermore, they conjectured that the constant 1 / 3 could be improved to 1 / 4 for a sufficiently large n . Their conjecture remains unsettled. In the present paper, it is proved that if G is a Hamiltonian plane triangulation with n vertices and minimum degree at least 4, then γ ( G ) ź max { ź 2 n / 7 ź , ź 5 n / 16 ź } . It follows immediately that if G is a 4-connected plane triangulation with n vertices, then γ ( G ) ź max { ź 2 n / 7 ź , ź 5 n / 16 ź } .