Kőnig's line colouring theorem (${\chi }^{\prime }(G)=\Delta (G)$)

For every finite bipartite graph, the edge chromatic number equals its maximum degree. Repeatedly remove a matching covering maximum-degree vertices in a bipartite graph: each removal reduces the maximum degree by at least one. The removed matchings in a graph are colour classes of a proper edge colouring. The reverse bound follows from the distinct colours required at a vertex of maximum degree.