Looseness ranges of triangulations on closed surfaces

The looseness (G) of a triangulation G on a closed surface F 2 is defined as the minimum number k such that for any surjection c : V (G) → {1, 2, . . . , 3+ k}, there exists a face uvw of G which gets three distinct colors c(u), c(v) and c(w). We define min(G) and max(G) as the minimum and the maximum of (G′) taken over all triangulationsG′ on F 2 isomorphic to G as graphs. We shall show that max(G)− min(G) 2 (2− (F 2))/2 , where (F 2) stands for the Euler characteristic (F 2), and in particular that two triangulations on the projective plane have the same looseness if they are isomorphic as graphs. © 2005 Elsevier B.V. All rights reserved.