Looseness ranges of triangulations on closed surfaces

It is shown that two triangulations on the projective plane have the same looseness if they are isomorphic as graphs.

Abstract

The looseness ξ(G) of a triangulation G on a closed surface F2 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 ξmin(G) and ξmax(G) as the minimum and the maximum of ξ(G′) taken over all triangulations G′ on F2 isomorphic to G as graphs. We shall show that ξmax(G)-ξmin(G)⩽2⌊(2-χ(F2))/2⌋, where χ(F2) stands for the Euler characteristic χ(F2), and in particular that two triangulations on the projective plane have the same looseness if they are isomorphic as graphs.