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 triangle- and quadrangle-free (353) configurations. 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 first known example of a 10-cage (Rev. Roumaine Math. Pure Appl. 18 (1973) 1033-1043), and the corresponding Balaban configuration. It is shown that the latter is linear, that is, it can be realized as a geometric configuration of points and lines in the Euclidean plane. Finally, based on the Balaban configuration, an infinite series of linear triangle-free and quadrangle-free ((7n)3) configurations is produced for each odd integer n?5.

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