State and prove Vizing's theorem about the chromatic index $\chi^{\prime}(G)$ of a graph $G$.

Let $K_{m, n}$ be the complete bipartite graph with class sizes $m$ and $n$. By first considering $\chi^{\prime}\left(K_{n, n}\right)$, find $\chi^{\prime}\left(K_{m, n}\right)$ for all $m$ and $n$.

Let $G$ be the graph of order $2 n+1$ obtained by subdividing a single edge of $K_{n, n}$ by a new vertex. Show that $\chi^{\prime}(G)=\Delta(G)+1$, where $\Delta(G)$ is the maximum degree of $G$.