Define the maximum degree $\Delta(G)$ and the chromatic index $\chi^{\prime}(G)$ of the graph $G$.

State and prove Vizing's theorem relating $\Delta(G)$ and $\chi^{\prime}(G)$.

Let $G$ be a connected graph such that $\chi^{\prime}(G)=\Delta(G)+1$ but, for every subgraph $H$ of $G, \chi^{\prime}(H)=\Delta(H)$ holds. Show that $G$ is a circuit of odd length.