The name "Johnson skeleton graphs" is used for the skeletons of the Johnson solids, which are polyhedral graphs.
Special cases are summarized in the following table.
The Johnson skeleton graphs 
and 
are minimal forbidden graphs
for unit-distance graphs.
The Johnson skeleton graphs 
, 
, 
, and 
contain induced subgraphs
admitting partial proper colorings using four
colors that tangle the Kempe chains in the original
two-interchange argument of Kempe's coloring
algorithm. They have 9, 10, 16, and 16 vertices,
respectively, and are the skeletons of the gyroelongated
square pyramid, gyroelongated square
dipyramid, snub square antiprism, and
disphenocingulum. The Fritsch
graph, corresponding to 
, is another such example. Note that these obstructions
to Kempe's argument do not imply that the Kempe-Kittell
coloring algorithm fails.
The skeleton of the gyroelongated pentagonal pyramid 
appeared in Zaks (1976) and was used by Owens (1980)
in the construction of a 76-node polyhedral quintic nonhamiltonian graph (though
neither author identified the graph as the skeleton
of a particular polyhedron).
An unrelated family of graphs known as Johnson graphs 
is defined as graphs whose vertices
are given by the k-subsets of 
, with two vertices adjacent iff their intersection
has size 
.
The Johnson skeleton graph 
is an outer-cylindrical forbidden minor 
(No. 36; Archdeacon et al. 2001).
