Chromatic index of a graph with vertices of degree 3 and one of degree 2

I would like to prove that the chromatic index of a graph with vertices of degree 3 and one vertex of degree 2 is 4.

I know:

• That this graph is in fact 3-regular graph (cubic graph) with one edge subdivided.
• That this graph has odd number of vertices.
• That its chromatic index is 3 or 4 (from Vizing's theorem), so I it would suffice to prove that its chromatic index can't be 3.

Suppose you have a proper $3$-coloring of the edges: each vertex is incident with a red edge, a blue edge, and a green edge, except the vertex of degree $2$, which is incident with a red edge and a blue edge. Then the degree-sum of the red subgraph is $n$ (the number of vertices), while the degree-sum of the green subgraph is $n-1$. This is impossible, because both degree-sums have to be even.