Colouring graph's edges. Let $G$ be a graph in which each vertex except one has degree $d$. Show that if $G$ can be edge-coloured in $d$ colours then
(1) $G$ has an odd number of vertices,
(2) $G$ has a vertex of degree zero
Please help me with it.

Hint:


*

*Let $u$ be the vertex of non-$d$ degree. We have that $\deg(u) < d$, so there is a color $c$ not used by that vertex.

*Remove all the edges of color $c$, either

*

*$\deg(u)$ is still smaller and we reduced the problem, or 

*all the degrees are equal, but the sum of degrees we removed is an even number.



I hope this helps $\ddot\smile$
