Extending the dual of the Petersen graph
This paper concerns a special class of residually connected geometries of rank 3 with flag-transitive automorphism group. Such a geometry can be described by a diagram in which the types of the rank 2 residues are listed. The class considered here is defined by the following two types of residues. One is a circle geometry in the sense of \textit{F. Buekenhout} [J. Comb. Theory, Ser. A 27, 121-151 (1979; Zbl 0419.51003)], this is the geometry of vertices and edges of a finite complete graph. The other is the geometry dual to that of the Petersen graph. It is shown that each geometry of this class is a quotient of the universal 2-cover of one of the two examples which are described explicitely. The second example should be modified in the following way: lines are the pairs \((l,\pi)\) consisting of a \(2\)-set \(l\) of \(\Psi \) and a partition \(\pi =\{l,m,n\}\) of \(\Psi \) into three pairwise disjoint \(2\)-sets, one of which is equal to \(l\). A plane \(p\) is defined to be incident with the line \((l,\pi)\) if \(p=l\cup m\) or \(p=l\cup n\).
scientific article; zbMATH DE number 1266850