  In 1996, Tarjan and Matheson proved that if $G$ is a plane triangulated disc
with $n$ vertices, $\gamma (G)\le n/3$, where $\gamma (G)$ denotes the
domination number of $G$. Furthermore, they conjectured that the constant $1/3$
could be improved to $1/4$ for sufficiently large $n$. Their conjecture remains
unsettled.
  In the present paper, it is proved that if $G$ is a hamiltonian plane
triangulation with $|V(G)|=n$ vertices and minimum degree at least 4, then
$\gamma (G)\le\max\{\lceil 2n/7\rceil, \lfloor 5n/16\rfloor\}$. It follows
immediately that if $G$ is a 4-connected plane triangulation with $n$ vertices,
then $\gamma (G)\le\max\{\lceil 2n/7\rceil, \lfloor 5n/16\rfloor\} $. It then
follows that if $n\ge 26$, then $\gamma (G)\le \lfloor 5n/16\rfloor$.
