The Franklin graph is the cubic graph with 12 vertices shown above in a number of drawings in the plane. Its embedding on the Klein bottle divides it into faces having a minimal map coloring using six colors, thus providing the sole counterexample to the Heawood conjecture.
The Franklin 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 Franklin graph is also its own graph distance
graph with distance 3.
The torus graph embedding of the Franklin graph and a portion of its periodic planar graph cover are shown above.
A graph coloring of the Franklin graph using the fewest colors for its vertices is illustrated above.
The Franklin graph is nonplanar but Hamiltonian. It has LCF notations ![[5,-5]^6](/images/equations/FranklinGraph/Inline3.svg)
and ![[-5,-3,3,5]^3](/images/equations/FranklinGraph/Inline4.svg)
. The graph spectrum
of the Franklin graph is 
.
The Franklin graph is a toroidal graph, as illustrated above. The left-hand drawing shows an embedding in a fundamental region whose paired boundary sides are identified to form a torus. The right-hand drawing shows a finite patch of the corresponding periodic lift, illustrating how edges continue across these boundaries and where corresponding vertices in different regions represent the same vertex on the torus.
