I'm trying to prove the following
A cubic 3-edge connected graph $G = (V, E)$ allows partitions $T_{i}\subset E$ such that $G\setminus T_{i}$ is 2-edge connected, for $i = 1,\ldots, 5$.
In other words, I want to find a 5-coloring of $G$ such that removing the edges of any one color yields a 2-edge connected subgraph.
I have tried showing the property on several graphs, including snarks and the Peterson graph and it works for those examples.
It is possible to prove this? Do you think that there exist potential counterexamples I can look at? Any thoughts would be greatly appreciated.
Thank you