Harries graph

In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3-regular, undirected graph with 70 vertices and 105 edges.^[1]

Harries graph
The Harries graph
Named after	W. Harries
Vertices	70
Edges	105
Radius	6
Diameter	6
Girth	10
Automorphisms	120 (S5)
Chromatic number	2
Chromatic index	3
Genus	9
Book thickness	3
Queue number	2
Properties	Cubic Cage Triangle-free Hamiltonian
Table of graphs and parameters

The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3-vertex-connected and 3-edge-connected, non-planar, cubic graph. It has book thickness 3 and queue number 2.^[2]

The characteristic polynomial of the Harries graph is

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

History

In 1972, A. T. Balaban published a (3-10)-cage graph, a cubic graph that has as few vertices as possible for girth 10.^[3] It was the first (3-10)-cage discovered but it was not unique.^[4]

The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong in 1980.^[5] There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph and the Harries–Wong graph.^[6] Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.

Gallery

• 
  The chromatic number of the Harries graph is 2.
• 
  The chromatic index of the Harries graph is 3.
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  Alternative drawing of the Harries graph.
• 
  Alternative drawing emphasizing the graph's 4 orbits.

References

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