# Tutte 12-cage

Tutte 12-cage
The Tutte 12-cage
Named after W. T. Tutte
Vertices 126
Edges 189
Diameter 6
Girth 12
Automorphisms 12096
Chromatic number 2
Chromatic index 3
Properties Cubic
Cage
Hamiltonian
Semi-symmetric
Bipartite

In the mathematical field of graph theory, the Tutte 12-cage or Benson graph[1] is a 3-regular graph with 126 vertices and 189 edges named after W. T. Tutte.[2]

The Tutte 12-cage is the unique (3-12)-cage (sequence A052453 in OEIS). It was discovered by C. T. Benson in 1966.[3] It has chromatic number 2 (bipartite), chromatic index 3, girth 12 (as a 12-cage) and diameter 6. Its crossing number is 170 and has been conjectured to be the smallest cubic graph with this crossing number.[4][5]

## Construction

The Tutte 12-cage is a cubic Hamiltonian 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 group Z/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]

The characteristic polynomial of the Tutte 12-cage is

$(x-3)x^{28}(x+3)(x^2-6)^{21}(x^2-2)^{27}.\$

It is the only graph with this characteristic polynomial; therefore, the 12-cage is determined by its spectrum.

## References

1. ^ a b c Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15 (2008).
2. ^
3. ^ Benson, C. T. "Minimal Regular Graphs of Girth 8 and 12." Canad. J. Math. 18, 1091–1094, 1966.
4. ^
5. ^ Pegg, E. T. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 2009.
6. ^ Polster, B. A Geometrical Picture Book. New York: Springer, p. 179, 1998.
7. ^ Balaban, A. T. "Trivalent Graphs of Girth Nine and Eleven and Relationships Among the Cages." Rev. Roumaine Math 18, 1033–1043, 1973.
8. ^ 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.)
9. ^ Conder, Marston; Malnič, Aleksander; Marušič, Dragan; Potočnik, Primož (2006), "A census of semisymmetric cubic graphs on up to 768 vertices", Journal of Algebraic Combinatorics 23: 255–294, doi:10.1007/s10801-006-7397-3.