Balaban 11-cage

Balaban 11-cage
The Balaban 11-cage
Named after	Alexandru T. Balaban
Vertices	112
Edges	168
Radius	6
Diameter	8
Girth	11
Automorphisms	64
Chromatic number	3
Chromatic index	3
Properties	Cubic Cage Hamiltonian
Table of graphs and parameters

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 Alexandru T. Balaban.^[1]

The Balaban 11-cage is the unique (3,11)-cage. It was discovered by Balaban in 1973.^[2] The uniqueness was proved by Brendan McKay and Wendy Myrvold in 2003.^[3]

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.^[4]

It has independence number 52,^[5] 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.

The characteristic polynomial of the Balaban 11-cage is:

${\displaystyle (x-3)x^{12}(x^{2}-6)^{5}(x^{2}-2)^{12}(x^{3}-x^{2}-4x+2)^{2}\cdot }$

${\displaystyle \cdot (x^{3}+x^{2}-6x-2)(x^{4}-x^{3}-6x^{2}+4x+4)^{4}\cdot }$

${\displaystyle \cdot (x^{5}+x^{4}-8x^{3}-6x^{2}+12x+4)^{8}}$.

The automorphism group of the Balaban 11-cage is of order 64.^[4]

• 
  The chromatic number of the Balaban 11-cage is 3.
• 
  The chromatic index of the Balaban 11-cage is 3.
• 
  Alternative drawing of the Balaban 11-cage.^[6]