Structural characterization of projective flexibility

A triangulation T of a fixed surface Z is called flexible if its graph G(T) has two or more labeled embeddings in 2;. We establish a structural characterization of flexible triangulations of the projective plane. (~) 1998 Elsevier Science B.V. All rights reserved By V(.) and F(.) we denote the sets of vertices and faces, respectively. A 3-cycle is a cycle of length 3. A triangulation T : G --* Z of a fixed closed surface S with a graph G is an embedding of G in 2; with every face bounded by a 3-cycle of G. Combinatorially, triangulation T is completely determined by its graph G = G(T) to- gether with the face set F(T). Two triangulations T, T t : G ~ S are called isomorphic" provided there is an isomorphism zr : T ~ T ~, i.e., a permutation of V(G) such that uvw E F(T) if and only if rc(u)zr(v)zr(w) E F(T'). Especially, an isomorphism T ~ T is called an automorphism of T. We say T and T ~ are equivalent, written T = T ~, if F(T) = F(Tt), i.e., any 3-cycle of G bounds a face either in both T and T' (not necessarily with the same orientation) or in neither; otherwise we say T and T' are distinct. Note that distinct (respectively, nonisomorphic) triangulations are distinguish- able in the vertex-labeled (vertex-unlabeled) sense and that distinct triangulations may be isomorphic. An important theorem of Whitney (6) implies that any triangulation T of the 2-sphere 2;o is combinatorially unique, i.e., F(T) is uniquely determined by G(T). For higher-genus surfaces, Whitney's theorem fails and there may exist two or 1 Research supported by RGC Competitive Earmarked Research Grants under HKUST 595/94P and HKUST 707/96P.