McGee graph

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title: "McGee graph" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["individual-graphs", "regular-graphs"] description: "Graph with 24 vertices and 36 edges" topic_path: "general/individual-graphs" source: "https://en.wikipedia.org/wiki/McGee_graph" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Graph with 24 vertices and 36 edges ::

::data[format=table title="infobox graph"]

Field	Value
name	McGee graph
image	[[Image: McGee graph hamiltonian.svg
image_caption	The McGee graph
namesake	W. F. McGee
vertices	24
edges	36
automorphisms	32
girth	7
diameter	4
radius	4
chromatic_number	3
chromatic_index	3
properties	Cubic
Cage
Hamiltonian
book thickness	3
::

| name = McGee graph | image = [[Image: McGee graph hamiltonian.svg|220px]] | image_caption = The McGee graph | namesake = W. F. McGee | vertices = 24 | edges = 36 | automorphisms = 32 | girth = 7 | diameter = 4 | radius = 4 | chromatic_number = 3 | chromatic_index = 3 | properties = Cubic Cage Hamiltonian |book thickness=3|queue number=2}} 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.

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First discovered by Sachs but unpublished, the graph is named after McGee who published the result in 1960.{{cite journal | last1=McGee | first1=W. F. | title=A Minimal Cubic Graph of Girth Seven | journal=Canadian Mathematical Bulletin | volume=3 | issue=2 | pages=149–152 | date=1960 | doi=10.4153/CMB-1960-018-1 | doi-access=free}} Then, the McGee graph was proven the unique (3,7)-cage by Tutte in 1966.{{cite journal | last1=Wong | first1=Pak-Ken | title=Cages—A Survey | journal=Journal of Graph Theory | volume=6 | pages=1–22 | date=1982 | doi=10.1002/jgt.3190060103}}

The McGee graph requires at least eight crossings in any drawing of it in the plane. It is one of three non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these three graphs is the generalized Petersen graph , also known as the Nauru graph.

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. The graph is 1-planar.{{citation | last = Pupyrev | first = Sergey | editor-first1 = Vida | editor-last1 = Dujmović | editor-first2 = Fabrizio | editor-last2 = Montecchiani | contribution = OOPS: Optimized One-Planarity Solver via SAT | title = Proc. 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025) | series = Leibniz International Proceedings in Informatics (LIPIcs) | year = 2025 | volume = 357 | pages = 14:1–14:19 | doi = 10.4230/LIPIcs.GD.2025.14 | isbn = 978-3-95977-403-1 | doi-access = free

Algebraic properties

The characteristic polynomial of the McGee graph is :x^3(x-3)(x-2)^3(x+1)^2(x+2)(x^2+x-4)(x^3+x^2-4x-2)^4.

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.{{cite journal | last1 = Jajcay | first1 = Robert | last2 = Širáň | first2 = Jozef | doi = 10.26493/1855-3974.124.06d | issue = 2 | journal = Ars Mathematica Contemporanea | pages = 375–384 | title = Small vertex-transitive graphs of given degree and girth | volume = 4 | year = 2011| doi-access = free

The automorphism group of the McGee graph, meaning its group of symmetries, has 32 elements. This group is isomorphic to the group of all affine transformations of \mathbb{Z}/8\mathbb{Z}, i.e., transformations of the form

:: x \mapsto ax + b

where a,b \in \mathbb{Z}/8\mathbb{Z} and a is invertible, so a = 1, 3, 5, 7. This is one of the two smallest possible group G with an outer automorphism that maps every element g \in G to an element conjugate to g. Peter A. Brooksbank and Matthew S. Mizuhara (2014). On groups with a class-preserving outer automorphism, Involve. Vol. 7, No. 2, 171–179. doi:10.2140/involve.2014.7.171 https://msp.org/involve/2014/7-2/p04.xhtml

Gallery

Image:McGee graph crossing number.svg|The crossing number of the McGee graph is 8. Image:McGee graph 3COL.svg |The chromatic number of the McGee graph is 3. Image:McGee graph 3color edge.svg|The chromatic index of the McGee graph is 3. Image:Acyclic_coloring.svg|The acyclic chromatic number of the McGee graph is 3. Image:McGee graph.svg|Alternative drawing of the McGee graph.

References

References

1. "McGee Graph".
2. Kárteszi, F. "Piani finit ciclici come risoluzioni di un certo problemo di minimo." Boll. Un. Mat. Ital. 15, 522-528, 1960
3. Tutte, W. T. Connectivity in Graphs. Toronto, Ontario: University of Toronto Press, 1966
4. Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 209, 1989
5. {{Cite OEIS
6. (2009). "Crossing number graphs". Mathematica Journal.
7. Jessica Wolz, ''Engineering Linear Layouts with SAT''. Master Thesis, University of Tübingen, 2018
8. John C. Baez, What algebraic structures are related to the McGee graph?, https://mathoverflow.net/q/215211

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