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The Petersen graphis an undirected graph with 10 vertices and 15 edges. Itis a small graph that serves as a useful example and counterexample for manyproblems in graph theory. The Petersen graph is named for JuliusPetersen, who in1898 constructed it to be the smallest bridgeless cubicgraph with nothree-edge-coloring. Although the graph is generally credited to Petersen, ithad in fact first appeared 12 years earlier, in 1886.
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DonaldKnuth statesthat the Petersen graph is 'a remarkable configuration that serves as acounterexample to many optimistic predictions about what might be true forgraphs in general.' The Petersen graph is the complement of the linegraph of K 5.It is also the Kneser graph KG 5,2; this means that it has one vertexfor each 2-element subset of a 5-element set, and two vertices are connected byan edge if and only if the corresponding 2-element subsets are disjoint fromeach other.
As a Kneser graph of the form KG 2 n − 1, n− 1 it is an example of an oddgraph. Geometrically,the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a dodecahedron with opposite points, lines andfaces identified together.
The Petersen graph is nonplanar. Any nonplanar graph has as minors either the completegraph K 5,or the complete bipartite graph K 3,3, but the Petersen graphhas both as minors. The K 5 minor can be formed by contractingthe edges of a perfect matching, for instance the five short edges in the firstpicture. The K 3,3 minor can be formed by deleting one vertex(for instance the central vertex of the 3-symmetric drawing) and contracting anedge incident to each neighbor of the deleted vertex.
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Thesimplest non-orientable surface on which the Petersen graph can beembedded without crossings is the projectiveplane. This isthe embedding given by the hemi-dodecahedron construction of the Petersen graph.The projective plane embedding can also be formed from the standard pentagonaldrawing of the Petersen graph by placing a cross-cap within the five-point star at thecenter of the drawing, and routing the star edges through this cross-cap; theresulting drawing has six pentagonal faces. This construction forms a regular map and shows that the Petersen graphhas non-orientable genus 1. The automorphism group of the Petersen graph is the symmetricgroup S 5;the action of S 5 on the Petersen graph follows from itsconstruction as a Kneser graph. Every homomorphism of the Petersen graph to itselfthat doesn't identify adjacent vertices is an automorphism.
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As shown in thefigures, the drawings of the Petersen graph may exhibit five-way or three-waysymmetry, but it is not possible to draw the Petersen graph in the plane insuch a way that the drawing exhibits the full symmetry group of the graph. Despiteits high degree of symmetry, the Petersen graph is not a Cayleygraph. It isthe smallest vertex-transitive graph that is not a Cayley graph.