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

Abstract: We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned 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. We prove that for every integer $d$, every multigraph $G$ with maximum degree $\Delta$ is $(\lceil \frac{\Delta}{d} \rceil, d)$-edge colourable if $d$ is even 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 generalize several classic results on the chromatic index of a graph by Shannon, Vizing, Holyer, Leven and Galil.