The pentakis icosidodecahedral graph, illustrated above, is a graph on 42 vertices that is the skeleton of the pentakis icosidodecahedron.
It admits a partial proper coloring using four colors that tangles the Kempe chains in the original two-interchange argument of Kempe's coloring algorithm. This obstruction does not imply that the Kempe-Kittell coloring algorithm fails to color the graph.
The pentakis icosidodecahedral graph is implemented in the Wolfram Language as GraphData["PentakisIcosidodecahedralGraph"].
