The Franklin graph is the 12-vertex cubic graph shown above whose embedding on the Klein bottle divides it into regions having a minimal coloring using six colors, thus providing the sole counterexample to the Heawood conjecture. The graph is implemented in the Wolfram Language as GraphData["FranklinGraph"].
It is isomorphic to the 6-crossed prism graph, Knödel graph , and honeycomb toroidal graph .
The minimal coloring of the Franklin graph is illustrated above.
The Franklin graph is nonplanar but Hamiltonian. It has LCF notations and .
The graph spectrum of the Franklin graph is .