Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

Structural characterization of projective flexibility

A triangulation Upsilon of a fixed surface Sigma is called flexible if its graph G(T) has two or more labeled embeddings in Sigma. We establish a structural characterization of flexible triangulations of the projective plane. (C) 1998 Elsevier Science B.V. All rights reserved

Weinberg bounds over nonspherical graphs

Let Aut(G) and E(G) denote the automorphism group and the edge set of a graph G, respectively. Weinberg's Theorem states that 4 is a constant sharp upper bound on the ratio \Aut(G)\/\E(G)\ over planar (or spherical) 3-connected graphs G. We have obtained various analogues of this theorem for nonspherical graphs, introducing two Weinberg-type bounds for an arbitrary closed surface Sigma, namely: W-P(Sigma) and W-T (Sigma) (=) (def) (G) (sup) \Aut(G)\/\E(G)\, where supremum is taken over the polyhedral graphs G with respect to C for W-P(Sigma) and over the graphs G triangulating Sigma for W-T (Sigma). We have proved that Weinberg bounds are finite for any surface; in particular: W-P = W-T = 48 for the projective plane, and W-T = 240 for the torus. We have also proved that the original Weinberg bound of 4 holds over the graphs G triangulating the projective plane with at least 8 vertices and, in general, for the graphs of sufficiently large order triangulating a fixed closed surface Sigma. (C) 2000 John Wiley & Sons, Inc.X112sciescopu

Weinberg bounds over nonspherical graphs

Let Aut(G) and E(G) denote the automorphism group and the edge set of a graph G, respectively. Weinberg's Theorem states that 4 is a constant sharp upper bound on the ratio \textbackslash{}Aut(G)\textbackslash{}/\textbackslash{}E(G)\textbackslas h{} over planar (or spherical) 3-connected graphs G. We have obtained various analogues of this theorem for nonspherical graphs, introducing two Weinberg-type bounds for an arbitrary closed surface Sigma, namely: W-P(Sigma) and W-T (Sigma) (=) (def) (G) (sup) \textbackslash{}Aut(G)\textbackslash{}/\textbackslash{}E(G)\textbackslas h{}, where supremum is taken over the polyhedral graphs G with respect to C for W-P(Sigma) and over the graphs G triangulating Sigma for W-T (Sigma). We have proved that Weinberg bounds are finite for any surface; in particular: W-P = W-T = 48 for the projective plane, and W-T = 240 for the torus. We have also proved that the original Weinberg bound of 4 holds over the graphs G triangulating the projective plane with at least 8 vertices and, in general, for the graphs of sufficiently large order triangulating a fixed closed surface Sigma. (C) 2000 John Wiley & Sons, Inc