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

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 χ'_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 Δ is (⌈Δ/d⌉, d)-edge colourable if d is even and (⌈3Δ - 1/3d - 1⌉, d)-edge colourable if d is odd and these bounds are tight. We also prove that for every simple graph G, χ'_d(G) ∈{⌈Δ/d⌉, ⌈Δ+1/d⌉} and characterize the values of d and Δ for which it is NP-complete to compute χ'_d(G). These results generalize several classic results on the chromatic index of a graph by Shannon, Vizing, Holyer, Leven and Galil.

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