Prove that the edges of the cubic graph G cannot be coloured with three colours such that adjacent edges have different colours.

Question

Let G be a connected cubic graph having a bridge $e = uv$

Prove that the edges of G cannot be coloured with three colours such that adjacent edges have different colours.

I started just by drawing the bridge between u and v and coloring it blue then drew two more edges incident with both u and v eventually showed it was true for a cubic graph on 4 vertices then moved onto the peterson graph and read there proof here Prove that the Petersen graph does not have edge chromatic number = 3. but not sure how to generalize this result to the question at hand.

Answer 1

Consider the subgraph formed by just two of the colors, including the color used on the bridge $uv$.

It is two-regular, meaning that it is a union of cycles. In particular, $uv$ is part of a cycle, contradicting the assumption that it is a bridge.