Geodesic
Looseness and Independence Number of Triangulations on Closed Surfaces
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 545-554

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The looseness of a triangulation G on a closed surface F^2, denoted by ξ (G), is defined as the minimum number k such that for any surjection c : V (G) →1, 2, . . ., k + 3, there is a face uvw of G with c(u), c(v) and c(w) all distinct. We shall bound ξ (G) for triangulations G on closed surfaces by the independence number of G denoted by α(G). In particular, for a triangulation G on the sphere, we have ξ (G) ≤11 α (G) - 10/6and this bound is sharp. For a triangulation G on a non-spherical surface F^2, we have ξ (G) ≤ 2 α (G) + 𝓁(F^2) − 2, where 𝓁(F^2) = (2 − χ (F^2))//2 with Euler characteristic χ (F^2).
Keywords: triangulations, closed surfaces, looseness, k-loosely tight, independence number 
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Nakamoto, Atsuhiro; Negami, Seiya; Ohba, Kyoji; Suzuki, Yusuke. Looseness and Independence Number of Triangulations on Closed Surfaces. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 545-554. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a2/