# Petersen coloring conjecture

This extrordainary conjecture asserts that in a very strong sense, every bridgeless cubic graph has all of the cycle-space properties posessed by the Petersen graph. If true, this conjecture would imply both The Berge-Fulkerson conjecture and The five cycle double cover conjecture.

If is a graph and we say that is a *binary cycle* if every vertex in the graph has even degree. If is a graph and is a map, we say that is *cycle-continuous* if the pre-image of every binary cycle is a binary cycle. The following conjecture is an equivalent reformulation of the Petersen coloring conjecture.

**Conjecture (Petersen coloring conjecture (2))**Every bridgeless graph has a cycle-continuous mapping to the Petersen graph.

### Re:

It is trivially true for all that are 3-edge-colorable -- which is the vast majority. Among the rest, I checked it using computer and lists of snarks for all graphs upto 34 vertices. (And some more -- e.g. all flower-snarks.)

Best wishes, Robert

## Question

For which bridgeless cubic graphs has this been checked for?