TAILIEUCHUNG - Báo cáo toán học: "Goldberg-Coxeter Construction for 3- and 4-valent Plane Graphs"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:Goldberg-Coxeter Construction for 3- and 4-valent Plane Graphs. | Goldberg-Coxeter Construction for 3- and 4-valent Plane Graphs Mathieu DUTOUR Michel DEZA LIGA ENS CNRS Paris and LIGA ENS CNRS Paris and Hebrew University Jerusalem Institute of Statistical Mathematics Tokyo Submitted Jun 13 2003 Accepted Jan 19 2004 Published Mar 5 2004 MR Subject Classifications Primary 52B05 52B10 52B15 Secondary 05C30 05C07 Abstract We consider the Goldberg-Coxeter construction GCkyl G0 a generalization of a simplicial subdivision of the dodecahedron considered in Gold37 and Cox71 which produces a plane graph from any 3- or 4-valent plane graph for integer parameters k l. A zigzag in a plane graph is a circuit of edges such that any two but no three consecutive edges belong to the same face a central circuit in a 4-valent plane graph G is a circuit of edges such that no two consecutive edges belong to the same face. We study the zigzag or central circuit structure of the resulting graph using the algebraic formalism of the moving group the k l -product and a finite index subgroup of SL2 Z whose elements preserve the above structure. We also study the intersection pattern of zigzags or central circuits of GCk l G0 and consider its projections obtained by removing all but one zigzags or central circuits . Key words. Plane graphs polyhedra zigzags central circuits. 1 Introduction As initial graph Go for the Goldberg-Coxeter construction we consider mainly i 3- and 4-valent 1-skeleton of Platonic and semiregular polyhedra prisms and antiprisms see Table 1 ii 3-valent graphs related to fullerenes and other chemically-relevant polyhedra iii 4-valent plane graphs which are minimal projections for some interesting alternating links those links are denoted according to Rolfsen s notation Rol76 see also for example Kaw96 . Research financed by EC s IHRP Programme within the Research Training Network Algebraic Combinatorics in Europe grant HPRN-CT-2001-00272. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R20 1 .

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