McGee graph

In the mathematical field of graph theory, the McGee graph or the (3-7)-cage is a 3-regular graph with 24 vertices and 36 edges.[1]

McGee graph
The McGee graph
Named afterW. F. McGee
Vertices24
Edges36
Radius4
Diameter4[1]
Girth7[1]
Automorphisms32[1]
Chromatic number3[1]
Chromatic index3[1]
Book thickness3
Queue number2
PropertiesCubic
Cage
Hamiltonian
Table of graphs and parameters

The McGee graph is the unique (3,7)-cage (the smallest cubic graph of girth 7). It is also the smallest cubic cage that is not a Moore graph.

First discovered by Sachs but unpublished,[2] the graph is named after McGee who published the result in 1960.[3] Then, the McGee graph was proven the unique (3,7)-cage by Tutte in 1966.[4][5][6]

The McGee graph requires at least eight crossings in any drawing of it in the plane. It is one of five non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these five graphs is the generalized Petersen graph G(12,5), also known as the Nauru graph.[7][8]

The McGee graph has radius 4, diameter 4, chromatic number 3 and chromatic index 3. It is also a 3-vertex-connected and a 3-edge-connected graph. It has book thickness 3 and queue number 2.[9]

Algebraic properties

The characteristic polynomial of the McGee graph is

.

The automorphism group of the McGee graph is of order 32 and doesn't act transitively upon its vertices: there are two vertex orbits, of lengths 8 and 16. The McGee graph is the smallest cubic cage that is not a vertex-transitive graph.[10]

References
  1. Weisstein, Eric W. "McGee Graph". MathWorld.
  2. Kárteszi, F. "Piani finit ciclici come risoluzioni di un certo problemo di minimo." Boll. Un. Mat. Ital. 15, 522-528, 1960
  3. McGee, W. F. "A Minimal Cubic Graph of Girth Seven." Canad. Math. Bull. 3, 149-152, 1960
  4. Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966
  5. Wong, P. K. "Cages--A Survey." J. Graph Th. 6, 1-22, 1982
  6. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 209, 1989
  7. Sloane, N. J. A. (ed.). "Sequence A110507 (Number of nodes in the smallest cubic graph with crossing number n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. Pegg, E. T.; Exoo, G. (2009), "Crossing number graphs", Mathematica Journal, 11.
  9. Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
  10. Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
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