Graph theory: Prove $k$-regular graph $\#V$ = odd, $\chi'(G)> k$

I'm looking to prove that any $k$-regular graph $G$ (i.e. a graph with degree $k$ for all vertices) with an odd number of points has edge-colouring number $>k$ ($\chi'(G) > k$).

With Vizing, I see that $\chi'(G) \leq k + 1$, so apparently $\chi'(G)$ will end up equaling $k+1$.

Furthermore, as $\#V$ is odd, $k$ must be even for $\#V\cdot k$ to be an even number (required to be even, since $\#V\cdot k = \frac{1}{2} \cdot \#E$.

Does anyone have any suggestions on what to try?

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Let $G=\langle V,E\rangle$ be a $k$-regular graph with $n=2m+1$ vertices; as you say, clearly $k=2\ell$ for some $\ell$, so $G$ has $$\frac{kn}2=\frac{2\ell(2m+1)}2=\ell(2m+1)$$ edges. Suppose that $c:E\to\{1,\dots,k\}$ is a coloring of the edges of $G$. $$\frac{\ell(2m+1)}k=m+\frac12\;,$$ so there is some color that is used on at least $m+1$ edges. $G$ has only $2m+1$ vertices, so two of these edges must share a vertex, and $c$ therefore cannot be a proper coloring.