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If $G$ is a regular graph with an uneven number of vertices, prove that $\chi'(G) = \Delta(G) + 1$;
where $\Delta(G)$ is the maximum degree (in this case any degree) of $G$ and $\chi'(G)$ is the chromatic index (edge chromatic number).
The only other option is, that $\chi'(G)=\Delta(G)$.
But then every (edge-)color class constitutes a perfect matching, which implies that the number of vertices must be even.