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We call a multigraph $(k,d)$-edge colourable if its edge set can bepartitioned into $k$ subgraphs of maximum degree at most $d$ and denote as$\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. Weprove that for every integer $d$, every multigraph $G$ with maximum degree$\Delta$ is $(\lceil \frac{\Delta}{d} \rceil, d)$-edge colourable if $d$ iseven and $(\lceil \frac{3\Delta - 1}{3d - 1} \rceil, d)$-edge colourable if $d$is odd and these bounds are tight. We also prove that for every simple graph$G$, $\chi'_{d}(G) \in \{ \lceil \frac{\Delta}{d} \rceil, \lceil\frac{\Delta+1}{d} \rceil \}$ and characterize the values of $d$ and $\Delta$for which it is NP-complete to compute $\chi'_d(G)$. These results generalizeseveral classic results on the chromatic index of a graph by Shannon, Vizing,Holyer, Leven and Galil.

Vizing's and Shannon's Theorems for defective edge colouring