The Franklin graph is named after Philip Franklin, who disproved the Heawood conjecture on the number of colors needed when a two-dimensional surface is partitioned into cells by a graph embedding.

[1] The Heawood conjecture implied that the maximum chromatic number of a map on the Klein bottle should be seven, but Franklin proved that in this case six colors always suffice.

(The Klein bottle is the only surface for which the Heawood conjecture fails.)

The Franklin graph can be embedded in the Klein bottle so that it forms a map requiring six colors, showing that six colors are sometimes necessary in this case.

It acts transitively on the vertices of the graph, making it vertex-transitive.