 Solution

Use mathematical induction on the maximum degree $\Delta$, including $\Delta =0$ as the base case. A graph of maximum degree zero has no edges, so zero colours suffice.

For $\Delta >0$, part (ii) supplies a matching in a graph $M$ meeting every vertex of vertex degree $\Delta$. Delete its edges, retaining all vertices. Each vertex formerly of maximum degree loses one incident edge; every other vertex already had vertex degree at most $\Delta -1$. HenceThe resulting graph is still a bipartite graph. By mathematical induction, it has a proper edge colouring using at most $\Delta -1$ colours. Give every edge of $M$ one new colour. Since $M$ is a matching in a graph, this produces a proper edge colouring of $G$ with at most $\Delta$ colours. The edges at a vertex of maximum degree require at least $\Delta$ colours, soThis is Kőnig's line colouring theorem.