The Frucht graph is a graph on 12 vertices and 18 edges that is one of the five smallest cubic identity graphs, of which two are planar (cf. Skiena 1990, p. 185). It is illustrated above in a number of drawings.
The Frucht graph is implemented in the Wolfram Language as GraphData["FruchtGraph"].
The Frucht graph is Hamiltonian and unit-distance.
It has three inequivalent order-1 LCF notations: [
, 
, 
,
2, 5, 
,
2, 5, 
,
, 4, 2], [
, 
,
2, 3, 
,
4, 
, 5, 2, 
, 
,
2], and [
,
2, 
, 
, 2, 3, 
, 5, 
, 2, 4, 
].
The cubic symmetric graph 
, which was the first known cubic
1-arc-transitive graph, is another graph
associated with Frucht (Frucht 1952).
