About a Brooks-type theorem for improper colouring

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A graph is k-improperly -colourable if its vertices can be partitioned into parts such that each part induces a subgraph of maximum degree at most k. A result of Lovasz a states that for any graph G, such a partition exists if l is at least (Δ(G)+1)/(k+1) . When k = 0, this bound can be reduced by Brooks' Theorem, unless G is complete or an odd cycle. We study the following question, which can be seen as a generalisation of the celebrated Brooks' Theorem to improper colouring: does there exist a polynomial-time algorithm that decides whether a graph G of maximum degree Δ has k-improper chromatic number at most (Δ+1)/(k+1) - 1? We show that the answer is no, unless P = N P , when Δ= (k + 1)l, k>0 and l+√l < 2k + 4. We also show that, if G is planar, k = 1 or k = 2, Δ = 2k + 2, and l= 2, then the answer is still no, unless P = N P . These results answer some questions of Cowen et al. [Journal of Graph Theory 24(3):205-219, 1997].

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