Partitioning into degenerate graphs in linear time

04/23/2022

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by   Thimothée Corsini, et al.

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Let G be a connected graph with maximum degree Δ≥ 3 distinct from K_Δ + 1. Generalizing Brooks' Theorem, Borodin, Kostochka and Toft proved that if p_1, …, p_s are non-negative integers such that p_1 + … + p_s ≥Δ - s, then G admits a vertex partition into parts A_1, …, A_s such that, for 1 ≤ i ≤ s, G[A_i] is p_i-degenerate. Here we show that such a partition can be performed in linear time. This generalizes previous results that treated subcases of a conjecture of Abu-Khzam, Feghali and Heggernes <cit.>, which our result settles in full.