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.
Best Answer
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.