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Their conjecture remains unsettled.\n  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 $\\gamma (G)\\le\\max\\{\\lceil 2n/7\\rceil, \\lfloor 5n/16\\rfloor\\}$. 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Plummer, Xiaoya Zha","submitted_at":"2014-08-20T06:04:24Z","abstract_excerpt":"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.\n  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 $\\gamma (G)\\le\\max\\{\\lceil 2n/7\\rceil, \\lfloor 5n/16\\rfloor\\}$. 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