what is the chromatic index of this graph

Question

I am trying to figure out the chromatic index of this graph. I thought that it is 4, however in the solutions that I have it says that the chromatic index is only 3. Which is the correct answer?

Answer 1

The chromatic index of this graph is indeed $4$. There are $4$ edges incident on vertex $1$. So none of them can have the same color, so we have used up $4$ colors. The other portion of the graph does not pose any further problems.

Also see Vizig's theorem.

Answer 2

By Vizing's theorem, the chromatic index of a graph is always either $\Delta$ or $\Delta+1$, where $\Delta$ denotes the maximum degree of the graph. It is well-known that if $G$ is bipartite, its chromatic index is exactly $\Delta$. Your graph is a tree, which is bipartite. Thus, its chromatic index is $\Delta = 4$.