Editing Tutte–Coxeter graph (section)

== Constructions and automorphisms ==
The Tutte–Coxeter graph is the [[bipartite graph|bipartite]] [[Levi graph]] connecting the 15 [[perfect matching]]s of a 6-vertex [[complete graph]] K<sub>6</sub> to its 15 edges, as described by Coxeter (1958b), based on work by Sylvester (1844). Each vertex corresponds to an edge or a perfect matching, and connected vertices represent the [[incidence structure]] between edges and matchings.

Based on this construction, Coxeter showed that the Tutte–Coxeter graph is a [[symmetric graph]]; it has a [[group (mathematics)|group]] of 1440 [[graph automorphism|automorphisms]], which may be identified with the automorphisms of the group of permutations on six elements (Coxeter 1958b). The [[inner automorphism]]s of this group correspond to permuting the six vertices of the ''K''<sub>6</sub> graph; these permutations act on the Tutte–Coxeter graph by permuting the vertices on each side of its bipartition while keeping each of the two sides fixed as a set. In addition, the [[outer automorphism]]s of the group of permutations swap one side of the bipartition for the other. As Coxeter showed, any path of up to five edges in the Tutte–Coxeter graph is equivalent to any other such path by one such automorphism.