Vizing theorem

Every finite simple graph of maximum degree $\Delta$ has edge chromatic number $\Delta$ or $\Delta +1$.

In a partial proper edge colouring with one uncoloured edge incident with a vertex $x$, a Vizing fan is a sequence of neighbours of $x$ whose incident edge colours are missing at earlier fan vertices. With one more colour than the maximum degree, fan rotations and a two-colour Kempe chain free one common colour at the endpoints of the uncoloured edge.