On extensions of partial geometries containing small \(\mu\)-subgraphs. (Russian) Zbl 0934.51005

Makhnev, A. A.

Diskretn. Anal. Issled. Oper. 3, No. 3, 71-83 (1996).

The author generalizes the results obtained by D. R. Hughes [Eur. J. Comb. 11, No. 5, 459-471 (1990; Zbl 0719.51005)] and A. Del Fra, D. Ghinelli-Smit, and D. R. Hughes [Geom. Dedicata 42, No. 2, 119-128 (1992; Zbl 0759.51004)].
In this paper there is proved that a geometry, which is a \(\varphi\)-homogeneous extension of partial geometry \(EpG_\alpha(s,t)\), \(\varphi\leq s\) containing \((a,B)\)-antiflag with \(\mu(a,b) =\mu (a,c)=\varphi(1+{t(\varphi-1)\over\alpha})\); \(b,c\in B\), is one of the following geometries: triangular extension or 6-homogeneous extension of generalized quadrangle, extension of a net or dual 2-block design. The \(\varphi\)-homogeneous extensions of partial geometries with strongly regular point graph, where \(\mu\) is given above, are classified.

Reviewer: E.Brožíková (Praha)

MSC:

51E14	Finite partial geometries (general), nets, partial spreads
05B25	Combinatorial aspects of finite geometries

Keywords:

\(\varphi\)-homogeneous geometries

Citations:

Zbl 0719.51005; Zbl 0759.51004