Harries–Wong graph
{{infobox graph
| name = Harries–Wong graph
| image = 220px
| image_caption = The Harries–Wong graph
| namesake = W. Harries,
Pak-Ken Wong
| vertices = 70
| edges = 105
| automorphisms = 24 (S4)
| girth = 10
| genus = 9
| diameter = 6
| radius = 6
| chromatic_number = 2
| chromatic_index = 3
| properties = Cubic
Cage
Triangle-free
Hamiltonian
|book thickness=3|queue number=2}}
In the mathematical field of graph theory, the Harries–Wong graph is a 3-regular undirected graph with 70 vertices and 105 edges.{{MathWorld|urlname=Harries-WongGraph|title= Harries–Wong Graph}}
The Harries–Wong 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.Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
The characteristic polynomial of the Harries–Wong 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.A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1–5. 1972. It was the first (3-10)-cage discovered but it was not unique.Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [http://citeseer.ist.psu.edu/448980.html].
The complete list of (3-10)-cages and the proof of minimality was given by O'Keefe and Wong in 1980.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. There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph and the Harries–Wong graph.Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976. Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.
Gallery
Image:Harries-wong graph 2COL.svg|The chromatic number of the Harries–Wong graph is 2.
Image:Harries-wong graph 3color edge.svg|The chromatic index of the Harries–Wong graph is 3.
Image:harries-wong_graph_alternative_drawing.svg|Alternative drawing of the Harries–Wong graph.
Image:Harries-Wong graph - the 8 orbits.jpg|The 8 orbits of the Harries–Wong graph.