  We prove that every cyclically 4-edge-connected cubic graph that can be
embedded in the projective plane, with the single exception of the Petersen
graph, is 3-edge-colorable. In other words, the only (non-trivial) snark that
can be embedded in the projective plane is the Petersen graph.
  This implies that a 2-connected cubic (multi)graph that can be embedded in
the projective plane is not 3-edge-colorable if and only if it can be obtained
from the Petersen graph by replacing each vertex by a 2-edge-connected planar
cubic (multi)graph.
  This result is a nontrivial generalization of the Four Color Theorem, and its
proof requires a combination of extensive computer verification and
computer-free extension of existing proofs on colorability.
  An unexpected consequence of this result is a coloring-flow duality statement
for the projective plane: A cubic graph embedded in the projective plane is
3-edge-colorable if and only if its dual multigraph is 5-vertex-colorable.
Moreover, we show that a 2-edge connected graph embedded in the projective
plane admits a nowhere-zero 4-flow unless it is Peteren-like (in which case it
does not admit nowhere-zero 4-flows). This proves a strengthening of the Tutte
4-flow conjecture for graphs on the projective plane.
  Some of our proofs require extensive computer verification. The necessary
source codes, together with the input and output files and the complete set of
more than 6000 reducible configurations are available on Github
(https://github.com/edge-coloring) which can be considered as an Addendum to
this paper. Moreover, we provide pseudocodes for all our computer
verifications.
