Balaban 10-cage

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Balaban 10-cage
Balaban 10-cage.svg
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
Book thickness 3
Queue number 2
Properties Cubic

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


See also[edit]

Molecular graph


  1. ^ Weisstein, Eric W. "Balaban 10-Cage". MathWorld. 
  2. ^ Alexandru T. Balaban, A trivalent graph of girth ten, Journal of Combinatorial Theory Series B 12 (1972), 1–5.
  3. ^ Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [1].
  4. ^ Mary R. O'Keefe and Pak Ken Wong, A smallest graph of girth 10 and valency 3, Journal of Combinatorial Theory Series B 29 (1980), 91–105.
  5. ^ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
  6. ^ Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, Universität Tübingen, 2018