# Balaban 10-cage

Balaban 10-cage
The Balaban 10-cage
Named after Alexandru T. Balaban
Vertices 70
Edges 105
Diameter 6
Girth 10
Automorphisms 80
Chromatic number 2
Chromatic index 3
Book thickness 3
Queue number 2
Properties Cubic
Cage
Hamiltonian

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 (3,10)-cage discovered but it is not unique.[3]

The complete list of (3-10)-cages and the proof of minimality was given by Mary R. O'Keefe and Pak Ken Wong.[4] There exist 3 distinct (3-10)-cages, the other two being the Harries graph and the Harries–Wong graph.[5] Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.

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 a 3-edge-connected graph. The book thickness is 3 and the queue number is 2.[6]

The characteristic polynomial of the Balaban 10-cage is

${\displaystyle (x-3)(x-2)(x-1)^{8}x^{2}(x+1)^{8}(x+2)(x+3)(x^{2}-6)^{2}(x^{2}-5)^{4}(x^{2}-2)^{2}(x^{4}-6x^{2}+3)^{8}.}$