The 10-cages and derived configurations

Symmetry properties of the three 10-cages on 70 vertices are investigated. Being bipartite, these graphs are Levi graphs of triangleand quadrangle-free (353) con!gurations. For each of these graphs a Hamilton cycle is given via the associated LCF notation. Furthermore, the automorphism groups of respective orders 80, 120, and 24 are computed. A special emphasis is given to the Balaban 10-cage, the !rst known example of a 10-cage (Rev. Roumaine Math. Pure Appl. 18 (1973) 1033–1043), and the corresponding Balaban con*guration. It is shown that the latter is linear, that is, it can be realized as a geometric con!guration of points and lines in the Euclidean plane. Finally, based on the Balaban con!guration, an in!nite series of linear triangle-free and quadrangle-free ((7n)3) con!gurations is produced for each odd integer n¿ 5. c © 2003 Elsevier B.V. All rights reserved.