Looseness width of 5-connected triangulations on the torus

Let G be a triangulation on a closed surface and c : V ( G ) → { 1 , 2 , 3 , … , 3 + k } a color assignment of the vertices of G. Then a face uvw of G is said to be heterochromatic for c if its three corners u, v and w receive three distinct colors. Furthermore, G is said to be k-loosely tight if there is a heterochromatic face of G for any surjection c : V ( G ) → { 1 , 2 , 3 , … , 3 + k } . The looseness of G, denoted by ξ ( G ) , is defined as the minimum k such that G is k-loosely tight. We show that if G is 5-connected triangulation on the torus, then ξ ( G ) is independent of the embedding of G.