Balaban 10-cage

Named afterAlexandru T. Balaban

Vertices70

Edges105

Radius6

Balaban 10-cage
The Balaban 10-cage
Named after	Alexandru T. Balaban
Vertices	70
Edges	105
Radius	6
Diameter	6
Girth	10
Automorphisms	80
Chromatic number	2
Chromatic index	3
Genus	9
Book thickness	3
Queue number	2
Properties	Cubic Cage Hamiltonian
Table of graphs and parameters

In the mathematical field of graph theory, the Balaban 10-cage or Balaban (3,10)-cage is a 3-regular graph with 70 vertices and 105 edges named after Alexandru T. Balaban.^[1] Published in 1972,^[2] It was the first 10-cage discovered but it is not unique.^[3]

The proof of minimality of the number of vertices was given by Mary R. O'Keefe and Pak Ken Wong.^[4] There are 2 other distinct (3,10)-cages, the Harries graph and the Harries–Wong graph.^[5] The Harries–Wong graph and Harries graph are also cospectral.

The Balaban 10-cage has chromatic number 2, chromatic index 3, diameter 6, girth 10 and is hamiltonian. It is also a 3-vertex-connected graph and 3-edge-connected. The book thickness is 3 and the queue number is 2.^[6]

The characteristic polynomial of the Balaban 10-cage is

${\displaystyle (x-3)(x-2)(x-1)^{8}x^{2}(x+1)^{8}(x+2)(x+3)\cdot }$

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