On Some Flag-Transitive Extensions of the Dual Petersen Graph

We consider flag-transitive extensions of the dual Petersen graph satisfying one of the following "pathological" properties: (-riLL) there are pairs of collinear points incident to more than one line; ( -TT) there are triples of collinear points not incident to the same plane. We prove that ihere is just one example-satisfying (~LL), with six points and $6 as its au tomorphism group, and there are just four examples satisfying ( 7 T), with 18, 16, 32, and 32 points and groups 3 $ 6 , 2 4 . $ 5 , 2 5 ' $5, and 25 -As, respectively. The paper is organized as follows. In Section 1 we recall some known results and we explain some motivations of our investigation. The examples we will characterize are described in Section 2. The characterization theorems are stated in Section 3 and proved in Sections 5 and 6. In Section 4 we state some lemmas, to be used in Sections 5 and 6. © 1995 Academic Press, Inc.