The distinguishing numbers of graphs on closed surfaces

A graph G is said to be d -distinguishable if there is a labeling c : V ( G ) → { 1 , 2 , … , d } such that no automorphism of G other than the identity map preserves the labels of vertices given by c . The smallest d for which G is d -distinguishable is called the distinguishing number of G . We shall prove that every 4-representative triangulation on a closed surface, except the sphere, is 2-distinguishable after establishing a general theorem on the distinguishability of polyhedral graphs faithfully embedded on closed surfaces, and show that there is an upper bound for the distinguishing number of triangulations on a given closed surface, applying the re-embedding theory of triangulations.