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# The Petersen graph

Julius Petersen (1839-1910) was a Danish mathematician.
Around 1898 he constructed the graph bearing his name
as the smallest counterexample against the claim that
a connected bridgeless cubic graph has an edge colouring
with three colours.

## Construction

The Petersen graph is the complement of the
Johnson graph J(5,2).
Thus, it can be described as the graph with as vertex set
the pairs from a 5-set, where two pairs are joined when they
are disjoint. It is the unique strongly regular graph with
parameters *v* = 10, *k* = 3, λ = 0, μ = 1,
and has spectrum 3^{1} 1^{5} (-2)^{4}.
The Petersen graph is one of the Moore graphs (regular graphs of girth 5
with the largest possible number *k*^{2} + 1 of vertices).
Two other Moore graphs are known, namely the pentagon (*k* = 2)
and the Hoffman-Singleton graph
(*k* = 7). If there are other Moore graphs, they must have
valency 57 and 3250 vertices, but cannot have a transitive group.

The Petersen graph is also a cage
(graph with smallest possible number of vertices given its
valency and girth).

## Group

The full group of automorphisms is G = S_{5} acting rank 3,
with point stabilizer 2×S_{3}.

## Supergraphs

The Petersen graph is contained in the
complement of the Clebsch graph and the
Sp(4,2) Generalized Quadrangle and the
Hoffman-Singleton graph.
Its extended bipartite double is contained in the
Gewirtz graph.

## Subgraphs

Substructures belonging to the maximal subgroups of the automorphism group:
a) *A 4-coclique*.
There are 5 of these, forming a single orbit.
The stabilizer of one is S_{4}, with vertex orbit sizes 4+6.
The induced subgraph on the 6 is 3K_{2}.

b) *A split into two pentagons*.
There are 6 of these, forming a single orbit.
The stabilizer of one is 5:4, with vertex orbit size 10.
There are 12 pentagons, with stabilizer D_{10}.

c) *A vertex*.
There are 10 of these, forming a single orbit.
The stabilizer of one is 2×S_{3},
with vertex orbit sizes 1+3+6.

## Independence and chromatic number

The Petersen graph has independence number 4 and chromatic number 3.
The five independent sets of size 4 are the sets of four pairs on a
given symbol. The twenty 3-colorings are found by taking two
independent sets of size four (they have one vertex x in common)
and the remaining triple (the neighbours of x).

## Circuits and Hamiltonicity

The Petersen graph is maximally non-Hamiltonian:
there is a Hamiltonian path between any two nonadjacent vertices.
There are 12 pentagons, 10 hexagons, 0 heptagons, 15 octagons
20 nonagons and 0 decagons. The binary code spanned by the cycles
is a [15,6,5]-code. The 64 code words are the zero word,
the 12+10+15+20 = 57 cycles, and the 6 unions of two disjoint pentagons.

## References

D. A. Holton & J. Sheehan,
*The Petersen graph*,
Australian Mathematical Society Lecture Notes 7,
Cambridge University Press, 1993.
J. Petersen,
*Sur le théorème de Tait*,
L'Intermédiaire des Mathématiciens **5** (1898) 225-227.