The Tutte 12-cage is a cubicHamiltonian graph and can be defined by the LCF notation [17, 27, –13, –59, –35, 35, –11, 13, –53, 53, –27, 21, 57, 11, –21, –57, 59, –17]7.[6]
There are, up to isomorphism, precisely two generalized hexagons of order (2,2) as proved by Cohen and Tits. They are the split Cayley hexagon H(2) and its point-line dual. Clearly both of them have the same incidence graph, which is in fact isomorphic to the Tutte 12-cage.[1]
The Balaban 11-cage can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.[7]
Algebraic properties
The automorphism group of the Tutte 12-cage is of order 12,096 and is a semi-direct product of the projective special unitary group PSU(3,3) with the cyclic groupZ/2Z.[1] It acts transitively on its edges but not on its vertices, making it a semi-symmetric graph, a regular graph that is edge-transitive but not vertex-transitive. In fact, the automorphism group of the Tutte 12-cage preserves the bipartite parts and acts primitively on each part. Such graphs are called bi-primitive graphs and only five cubic bi-primitive graphs exist; they are named the Iofinova-Ivanov graphs and are of order 110, 126, 182, 506 and 990.[8]
All the cubic semi-symmetric graphs on up to 768 vertices are known. According to Conder, Malnič, Marušič and Potočnik, the Tutte 12-cage is the unique cubic semi-symmetric graph on 126 vertices and is the fifth smallest possible cubic semi-symmetric graph after the Gray graph, the Iofinova–Ivanov graph on 110 vertices, the Ljubljana graph and a graph on 120 vertices with girth 8.[9]
^Pegg, E. T. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 2009.
^Polster, B. A Geometrical Picture Book. New York: Springer, p. 179, 1998.
^Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math 18, 1033–1043, 1973.
^Iofinova, M. E. and Ivanov, A. A. "Bi-Primitive Cubic Graphs." In Investigations in the Algebraic Theory of Combinatorial Objects. pp. 123–134, 2002. (Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, pp. 137–152, 1985.)