How can we create a graph $G'$ by subdividing the edge $e$ with a new degree two vertex $z$?

Question

Let $G=(V,E)$ be a bipartite graph and e=uv an edge where $\text{deg}(u)<\Delta$. How can we create a graph G' by subdividing the edge e with a new degree two vertex z.

i.e: we delete $e$ and add a new vertex $z$ which is adjacent to $u$ and $v$.

Show that $X^E(G')=\Delta$.

$X^E(G')$ : the edge chromatic number of $G'$

Answer 1

I assume it is known that for the bipartite graph $G$, $X^E(G)=\Delta$. Consider an according edge colouring. Let $\alpha$ be the colour of $uv$. Since $\operatorname{deg}(u)<\Delta$, there is at least one color $\beta$ not occuring for an edge incident with $u$. Assign $\alpha$ to $zv$ and $\beta$ to $uz$.

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EDIT: It appears, we don't need that $G$ is bipartite. It is sufficient that $G$ is a class 1 graph and $\operatorname{deg}(u)<\Delta$.