| Petersen graph | |
|---|---|
 The Petersen graph is most commonly drawn as a pentagon with a pentagram inside, with five spokes. | |
| Named after | Julius Petersen |
| Vertices | 10 |
| Edges | 15 |
| Radius | 2 |
| Diameter | 2 |
| Girth | 5 |
| Automorphisms | 120 (S5) |
| Chromatic number | 3 |
| Chromatic index | 4 |
| Fractional chromatic index | 3 |
| Genus | 1 |
| Properties | Cubic Strongly regular Distance-transitive Snark |
| Table of graphs and parameters | |
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.[1][2]
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.[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.
- ^ Brouwer, Andries E., The Petersen graph
- ^ Petersen, Julius (1898), "Sur le théorème de Tait", L'Intermédiaire des Mathématiciens, 5: 225–227
- ^ Kempe, A. B. (1886), "A memoir on the theory of mathematical form", Philosophical Transactions of the Royal Society of London, 177: 1–70, doi:10.1098/rstl.1886.0002, S2CID 108716533
- ^ Knuth, Donald E., The Art of Computer Programming; volume 4, pre-fascicle 0A. A draft of section 7: Introduction to combinatorial searching