<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 6 Issue 3" %>
Near Polygons Having a big sub Near Polygon Isomorphic to
Bart De Bruyn
Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Gent, Belgium
bdb@cage.rug.ac.be
Annals of Combinatorics 6 (3) p.285-294 September, 2002
AMS Subject Classification: 05B25, 51E12, 51E25
Abstract:
In [1] a near 2n-gon ,, was constructed from the set of 2-factors of the complete graph on 2n+2 vertices. In this paper, we determine all near 2n-gons, , having as a big geodetically closed sub near 2(n-1)-gon under the additional assumption that every two points at distance 2 have at least two common neighbours. We will prove that such a near 2n-gon is either isomorphic to or to a direct product of with a line. As a corollary of that, the near polygon is characterized by means of the local space at one point.
Keywords: near polygon, generalized quadrangle

References:

1. A.E. Brouwer, A.M. Cohen, J.I. Hall, and H.A.Wilbrink, Near polygons and Fischer spaces, Geom. Dedicata 49 (1994) 349¨C368.

2. A.E. Brouwer and H.A. Wilbrink, The structure of near polygons with quads, Geom. Dedicata, 14 (1983) 145¨C176.

3. B. De Bruyn, Characterization of near hexagons by means of one local space, J. Combin. Theory, Ser. A, to appear.

4. S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Research Notes in Mathematics, Vol. 110, Pitman, Boston, 1984.

5. S.A. Shad and E.E. Shult, The near n-gon geometries, Unpublished, 1979.

6. E.E. Shult and A.Yanushka, Near n-gons and line systems, Geom. Dedicata 9 (1980) 1¨C72.

7. J. Tits, Sur la trialit¨¦ et certains groupes qui s¨¦n d¨¦duisent, Inst. Hautes Etudes Sci. Publ. Math. 2 (1959) 14¨C60.

8. H. Van Maldeghem, Generalized Polygons, Monographs in Mathematics, Vol. 93, Birkhäuser, Basel, Boston, Berlin, 1998.