Suppose you have a set with 5 elements. There are 10 ways to choose a 2-element subset. Form a graph with these 10 choices as vertices, and with two vertices connected by an edge precisely when the corresponding subsets are disjoint. You get the graph shown here, called the **Petersen graph**.

The picture above, made by Japheth Wood, also appears here:

• Japheth Wood, Proof without words: the automorphism group of the Petersen graph is isomorphic to \(\mathrm{S}_5\), *Mathematics Magazine* **89** (October 2016), 267.

As the title indicates, it’s easy to use this picture to determine the symmetry group of the Petersen graph.

The Petersen graph is reputed to be a counterexample to many conjectures about graph theory, and it shows up in many places. We have described it as an example of a ‘Kneser graph’. The **Kneser graph** \(KG_{n,k}\) is the graph whose vertices correspond to the \(k\)-element subsets of an \(n\)-element set, where two vertices are connected by an edge if and only if the two corresponding subsets are disjoint.

We can also get the Petersen graph by taking a regular dodecahedron and identifying antipodal vertices and edges.

Or, take the complete graph on 5 vertices, \(K_5\), and form a new graph with the edges of \(K_5\) as *vertices*, with two of these vertices connected by an edge if the corresponding edges in \(K_5\) do *not* share a vertex. The result is the Petersen graph! We say the Petersen graph is the complement of the line graph of \(K_5\).

For more details, see:

• Wikipedia, Petersen graph.

The Petersen graph also shows up when you consider all possible phylogenetic trees that could explain how some set of species arose from a common ancestor. These are binary trees with a fixed number of leaves where each edge is labelled by a time in $[0,\infty)$. The space of all such trees is contractible, but nonetheless topologically interesting. The space of phylogenetic trees with 4 leaves is the cartesian product of the cone on the Petersen graph and $[0,\infty)^5$. For pictures illustrating this, see:

• Louis Billera, Susan Holmes and Karen Vogtmann, Geometry of the space of phylogenetic trees, *Advances in Applied Mathematics* **27** (2001), 733-767.

For related ideas, see

• John Baez, Operads and the tree of life.

Here is another picture illustrating the relation between the Petersen graph and 2-element subsets of a 5-element set:

The picture here is adapted from one that Tilman Piesk created and put into the public domain on Wikicommons.

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The first picture also demonstrated the Petersen colouring, also known as normal 5-colouring.

(Every edge in a normally colored graph is normal, i.e. it uses together with its four neighbours either only three colors or all five colors.)