The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring.

[1][2] Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A.

Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration.

[3] Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general.

"[4] The Petersen graph also makes an appearance in tropical geometry.

The cone over the Petersen graph is naturally identified with the moduli space of five-pointed rational tropical curves.

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 and faces identified together.

The most common and symmetric plane drawing of the Petersen graph, as a pentagram within a pentagon, has five crossings.

Each edge in this drawing is crossed at most once, so the Petersen graph is 1-planar.

On a torus the Petersen graph can be drawn without edge crossings; it therefore has orientable genus 1.

The Petersen graph can also be drawn (with crossings) in the plane in such a way that all the edges have equal length.

The simplest non-orientable surface on which the Petersen graph can be embedded without crossings is the projective plane.

This is the embedding given by the hemi-dodecahedron construction of the Petersen graph (shown in the figure).

The projective plane embedding can also be formed from the standard pentagonal drawing of the Petersen graph by placing a cross-cap within the five-point star at the center of the drawing, and routing the star edges through this cross-cap; the resulting drawing has six pentagonal faces.

This construction forms a regular map and shows that the Petersen graph has non-orientable genus 1.

The Petersen graph is strongly regular (with signature srg(10,3,0,1)).

[8] As shown in the figures, the drawings of the Petersen graph may exhibit five-way or three-way symmetry, but it is not possible to draw the Petersen graph in the plane in such a way that the drawing exhibits the full symmetry group of the graph.

It is the smallest bridgeless cubic graph with no Hamiltonian cycle.

As a finite connected vertex-transitive graph that does not have a Hamiltonian cycle, the Petersen graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for a Hamiltonian path and is verified by the Petersen graph.

[9] To see that the Petersen graph has no Hamiltonian cycle, consider the edges in the cut disconnecting the inner 5-cycle from the outer one.

If there is a Hamiltonian cycle C, it must contain an even number of these edges.

Assume that the top edge of the cut is not contained in C (all the other cases are the same by symmetry).

The only remaining case is a Möbius ladder formed by connecting each pair of opposite vertices by a chord, which again has a 4-cycle.

Since the Petersen graph has girth five, it cannot be formed in this way and has no Hamiltonian cycle.

The snark theorem, a result conjectured by W. T. Tutte and announced in 2001 by Robertson, Sanders, Seymour, and Thomas,[10] states that every snark has the Petersen graph as a minor.

The long-standing Goldberg-Seymour Conjecture proposes that this is the largest gap possible.

The Thue number (a variant of the chromatic index) of the Petersen graph is 5.

The generalized Petersen graphs also include the n-prism

The Petersen family consists of the seven graphs that can be formed from the Petersen graph by zero or more applications of Δ-Y or Y-Δ transforms.