suppose a graph $G$ is 3-regular with bridge (cut edge),do we have $\chi^{'}(G)=4$?

Question

suppose a graph $G$ is 3-regular with bridge (cut edge),do we have $\chi^{'}(G)=4$?

I think that it is right but I couldn't prove it,can you give me some hint or guidance about it,thanks a lot.

Answer 1

Yes. Remove the bridge and let $H$ be one of the resulting components; I claim that $\chi'(H)=4$.

$H$ has one vertex of degree $2$ and $2n$ vertices of degree $3$, so $H$ has $3n+1$ edges. If the edges of $H$ are colored with $3$ colors, then there must be at least $\lceil\frac{3n+1}3\rceil=n+1$ edges of one color. Since edges of the same color can't be adjacent, that would require at least $2n+2$ vertices, but $H$ has only $2n+1$ vertices.