Structural characterization of projective flexibility

Chen, Beifang; Lawrencenko, Serge

doi:10.1016/s0012-365x(98)00052-1

Cited by 7 publications

(8 citation statements)

References 6 publications

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“…The structural characterization of flexible triangulations of the projective plane is established in [3]. In the present article, we reproduce the proof of this result for the sake of completeness.…”

Section: Lemma 10 the Action Of Any Automorphism Of A Triangulation mentioning

confidence: 65%

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Weinberg bounds over nonspherical graphs

2000

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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 Σ, namely:where supremum is taken over the polyhedral graphs G with respect to Σ for W P (Σ) and over the graphs G triangulating Σ for W T (Σ). 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

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Section: Lemma 10 the Action Of Any Automorphism Of A Triangulation mentioning

confidence: 65%

“…Lemma 12 (Chen and Lawrencenko [3]). All 1-flexible triangulations of the projective plane, up to isomorphisms, can be generated from the triangulations IT 1 [Fig.…”

Section: Lemma 10 the Action Of Any Automorphism Of A Triangulation mentioning

confidence: 99%

Weinberg bounds over nonspherical graphs

2000

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“…1(c); and the bunch of three bouquets,  BB, shaded in Fig. The structural characterization of flexible triangulations of the projective plane is established in [3]. In the present article, we reproduce the proof of this result for the sake of completeness.…”

Section: Itmentioning

confidence: 72%

“…This completes the proof. ■ Lemma (Chen and Lawrencenko [3]; Negami, Nakamoto and Tanuma [10]). There exists a constant upper bound on the number of flexible faces in a triangulation of a fixed surface.…”

Section:  Wmentioning

confidence: 99%

Weinberg bounds over nonspherical graphs

2000

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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 Σ, namely: where supremum is taken over the polyhedral graphs G with respect to Σ for WP(Σ) and over the graphs G triangulating Σ for WT(Σ). We have proved that Weinberg bounds are finite for any surface; in particular: WP = WT = 48 for the projective plane, and WT = 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 Σ. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 220–236, 2000

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“…Nakamoto and Ota [21] improved this bound to 171g − 72, which prior to this paper was the best known upper bound on the order of an irreducible triangulation of an arbitrary surface. In the case of orientable surfaces, Cheng et al [8] improved this bound to 120g. We prove: Theorem 1.…”

Section: Introductionmentioning

confidence: 99%