## Snarks on the projective plane

The Four-Color-Theorem is equivalent to the fact that every 3-connected cubic
planar graph is 3-edge-colorable. For any other surface S, there are snarks
(non-3-edge-colorable cyclically 4-edge-connected cubic graphs of girth at least
5) whose minimum genus embedding is in S. My student Andrej Vodopivec recently
proved that there are infinitely many nonisomorphic snarks that can be embedded
in the torus. This fact also implies that for every positive integer g there are
infinitely many snarks of genus g.

During the 21st LL-seminar in Celovec (Klagenfurt) in Austria, I asked the
following unresolved question:

**Problem:** Determine
all snarks that can be embedded in the projective plane.

I would not be surprised if the Petersen graph is the only projective snark.

Send comments to Bojan.Mohar@uni-lj.si

##### Revised: maj 11, 2004.