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 $\frac{1}{2}\cdot\#V\cdot k = \#E$.
Does anyone have any suggestions on what to try?