Balaban 11-cage

Balaban 11-cage
240px The Balaban 11-cage
Named after	A. T. Balaban
Vertices	112
Edges	168
Radius	6
Diameter	8
Girth	11
Automorphisms	64
Chromatic number	3
Chromatic index	3
Properties	Cubic Cage Hamiltonian

In the mathematical field of graph theory, the Balaban 11-cage or Balaban (3-11)-cage is a 3-regular graph with 112 vertices and 168 edges named after A. T. Balaban.

The Balaban 11-cage is the unique (3-11)-cage. It was discovered by Balaban in 1973. The uniqueness was proved by McKay and Myrvold in 2003.

The Balaban 11-cage is a Hamiltonian graph and can be constructed by excision from the Tutte 12-cage by removing a small subtree and suppressing the resulting vertices of degree two.

It has chromatic number 3, chromatic index 3, radius 6, diameter 8 and girth 11. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

Algebraic properties

The characteristic polynomial of the Balaban 11-cage is : $(x-3) x^{12} (x^2-6)^5 (x^2-2)^{12} (x^3-x^2-4 x+2)^2\cdot$ $\cdot(x^3+x^2-6 x-2) (x^4-x^3-6 x^2+4 x+4)^4 (x^5+x^4-8 x^3-6 x^2+12 x+4)^8$.

The automorphism group of the Balaban 11-cage is of order 64.

Gallery

• Balaban 11-cage 3COL.svg

  The chromatic number of the Balaban 11-cage is 3.

• Balaban 11-cage 3color edge.svg

  The chromatic index of the Balaban 11-cage is 3.

• Balaban 11-cage alternative drawing.svg

  Alternative drawing of the Balaban 11-cage.