The 10-cages and derived configurations

• Published in: Discrete mathematics Vol. 275; no. 1-3; pp. 265 - 276
• Main Authors: Pisanski, T., Boben, M., Marušič, D., Orbanić, A., Graovac, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 28.01.2004; Elsevier
• Subjects: Cages; Combinatorics; Combinatorics. Ordered structures; Configurations; Exact sciences and technology; Geometry; Graph theory; Graphs; Mathematics; Sciences and techniques of general use; Configurations; Graphs; Cages; Euclidean geometry; Cage; Configuration; Hamiltonian cycle; Incidence geometry; Discrete mathematics; Levi graph; Graph theory; Automorphism group
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/S0012-365X(03)00110-9

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.