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Dense Near Octagons with Four Points on Each Line, III
Bart De Bruyn
Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281 (S22), B-9000 Gent, Belgium
bdb@cage.ugent.be
Annals of Combinatorics 15 (1) pp.19-35 January, 2011
AMS Subject Classification: 05B25, 51E12
Abstract:
This is the third paper dealing with the classification of the dense near octagons of order (3; t). Using the artial classification of the valuations of the possible hexes obtained in [12], we are able to show that almost all such near octagons admit a big hex. Combining this with the results in [11], where we classified the dense near octagons of order (3; t) with a big hex, we get an incomplete classication for the dense near octagons of order (3; t): There are 28 known examples and a few open cases. For each open case, we have a rather detailed description of the structure of the near octagons involved.
Keywords: near polygon, near octagon, generalized quadrangle, valuation

References:

1. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer-Verlag, Berlin (1989)

2. Brouwer, A.E., Wilbrink, H.A.: The structure of near polygons with quads. Geom. Dedicata 14(2), 145–--176 (1983)

3. Cameron, P.J.: Dual polar spaces. Geom. Dedicata 12(1), 75–--85 (1982)

4. De Bruyn, B.: Near hexagons with four points on a line. Adv. Geom. 1(3), 211--–228 (2001)

5. De Bruyn, B.: The glueing of near polygons. Bull. Belg. Math. Soc. Simon Stevin 9(4), 621–--630 (2002)

6. De Bruyn, B.: Decomposable near polygons. Ann. Combin. 8(3), 251--–267 (2004)

7. De Bruyn, B.: Dense near polygons with two types of quads and three types of hexes. J. Combin. Math. Combin. Comput. 57, 113--–128 (2006)

8. De Bruyn, B.: Compatible spreads of symmetry in near polygons. J. Algebraic Combin. 23(2), 137--–148 (2006)

9. De Bruyn, B.: Near Polygons. Birkh¨auser Verlag, Basel (2006)

10. De Bruyn, B.: Valuations of glued near hexagons. J. Combin. Des. 15(1), 35--–48 (2007)

11. De Bruyn, B.: Dense near octagons with four points on each line, I. Ann. Combin. 11, 127--–142 (2007)

12. De Bruyn, B.: Dense near octagons with four points on each line, II. Adv. Geom. 7(2), 191--–206 (2007)

13. De Bruyn, B., Vandecasteele, P.: Valuations of near polygons. Glasg. Math. J. 47(2), 347--361 (2005)

14. De Bruyn, B., Vandecasteele, P.: The valuations of the near hexagons related to the Wittdesigns S(5;6;12) and S(5;8;24). J. Combin. Des. 14(3), 214–--228 (2006)

15. De Bruyn, B., Vandecasteele, P.: The classication of the slim dense near octagons. European J. Combin. 28(1), 410--–428 (2007)

16. Dixmier, S., Zara, F.: Etude d'un quadrangle g´en´eralis´e autour de deux de ses points non li´es. preprint (1976)

17. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles, Volume 110 of Research Notes in Mathematics. Pitman, Boston (1984)

18. Shad, S.A., Shult, E.E.: The near n-gon geometries. preprint (1979)

19. Shult, E.E., Yanushka, A.: Near n-gons and line systems. Geom. Dedicata 9(1), 1--–72 (1980)