Balaban 10-cage

Cubic graph with 70 nodes and 105 edges

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 complete list of 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, diameter6, 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

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

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

Gallery

• 
  The chromatic number of the Balaban 10-cage is 2.
• 
  The chromatic index of the Balaban 10-cage is 3.
• 
  Another drawing of the Balaban 10-cage.

See also

Molecular graph
Balaban 11-cage

References

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