The snub cubical graph is the Archimedean graph on 24 nodes and 60 edges obtained by taking the skeleton of the snub cube. It is a quintic graph, is planar, Hamiltonian, and has chromatic number 3. Several drawings are illustrated above.
The snub cubical graph also admits a partial four-coloring whose Kempe chains tangle under one order of the two interchanges in the faulty argument behind Kempe's coloring algorithm. This gives a counterexample to that step, but does not imply failure for the reverse interchange order.
It is implemented in the Wolfram Language as GraphData["SnubCubicalGraph"].
It is vertex-transitive, although not edge-transitive because some edges are part of graph triangles while others are not.
The snub cubical graph has 845,340 distinct undirected Hamiltonian cycles, giving two LCF-type notations of order 4 (illustrated above), 12 of order 3 (illustrated above), 627 of order 2, and 70127 of order 1.
Its graph spectrum is given by 
,
where 
,
, and 
are the polynomial roots
of 
.
Its automorphism group is of order 
.
