In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. 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.

Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A. B. Kempe (1886). 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.

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."

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.

The Petersen graph is the complement of the line graph of ${\displaystyle K_{5}}$. It is also the Kneser graph ${\displaystyle KG_{5,2}}$; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other. As a Kneser graph of the form ${\displaystyle KG_{2n-1,n-1}}$ it is an example of an odd graph.

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 Petersen graph is nonplanar. Any nonplanar graph has as minors either the complete graph ${\displaystyle K_{5}}$, or the complete bipartite graph ${\displaystyle K_{3,3}}$, but the Petersen graph has both as minors. The ${\displaystyle K_{5}}$ minor can be formed by contracting the edges of a perfect matching, for instance the five short edges in the first picture. The ${\displaystyle K_{3,3}}$ minor can be formed by deleting one vertex (for instance the central vertex of the 3-symmetric drawing) and contracting an edge incident to each neighbor of the deleted vertex.

The most common and symmetric plane drawing of the Petersen graph, as a pentagram within a pentagon, has five crossings. However, this is not the best drawing for minimizing crossings; there exists another drawing (shown in the figure) with only two crossings. Because it is nonplanar, it has at least one crossing in any drawing, and if a crossing edge is removed from any drawing it remains nonplanar and has another crossing; therefore, its crossing number is exactly 2. 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.