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
Vertices70
Edges105
Radius6
Diameter6
Girth10
Automorphisms120 (S5)
Chromatic number2
Chromatic index3
Book thickness3
Queue number2
PropertiesCubic
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

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.

  • The chromatic number of the Harries graph is 2.
  • The chromatic index of the Harries graph is 3.
  • 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.
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