Balaban 10-cage

Balaban 10-cage
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 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, 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

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

\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.

References

^ Weisstein, Eric W. "Balaban 10-Cage". MathWorld.
^ Alexandru T. Balaban, A trivalent graph of girth ten, Journal of Combinatorial Theory Series B 12 (1972), 1–5.
^ Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [1].
^ 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.
^ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
^ Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, Universität Tübingen, 2018

[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

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Gallery

See also

References