Four proofs of the directed Brooks' Theorem

Aboulker, P; Aubian, Guillaume

doi:10.48550/arxiv.2109.01600

Cited by 2 publications

(4 citation statements)

References 22 publications

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“…Brooks' Theorem follows from Theorem 2 by noting that a d-degenerate graph is (d + 1)-colorable. We should also mention that similar generalizations and variants of Brooks' Theorem exist: see for example [1] for a directed version or [9] for a distributed version.…”

Section: Introductionmentioning

confidence: 92%

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Partitioning into degenerate graphs in linear time

Corsini¹,

Quentin²,

Feghali³

et al. 2022

Preprint

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Let G be a connected graph with maximum degree ∆ ≥ 3 distinct from K∆+1. Generalizing Brooks' Theorem, Borodin, Kostochka and Toft proved that if p1, . . . , ps are non-negative integers such that p1 +• • • +ps ≥ ∆−s, then G admits a vertex partition into parts A1, . . . , As such that, for 1 ≤ i ≤ s, G[Ai] is pi-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 [2], which our result settles in full.

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Section: Introductionmentioning

confidence: 92%

“…, v dB +1 }, v n = z, and, similarly to the non-∆-regular case, the vertices of V \ (X ∪ {z}) are ordered in a post-order traversal of some tree T spanning G \ X rooted at z. We thus have the following analogue of Property (1).…”

Section: The Proof Of Theoremmentioning

confidence: 99%

Partitioning into degenerate graphs in linear time

Corsini¹,

Quentin²,

Feghali³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Harutyunyan and Mohar then gave a stronger result in [10]. Finally, Aboulker and Aubian gave four new proofs of the following theorem in [1].…”

Section: Graph (Re)colouringmentioning

confidence: 99%

“…Hence, one can wonder if Brooks' Theorem can be extended to digraphs using ∆ min (D) instead of ∆ max (D). Unfortunately, Aboulker and Aubian [1] proved that, given a digraph D, deciding whether D is ∆ min (D)-dicolourable is NP-complete. Thus, unless P=NP, we cannot expect an easy characterization of digraphs satisfying χ(D) = ∆ min (D) + 1.…”

Section: Graph (Re)colouringmentioning

confidence: 99%

Strengthening the Directed Brooks' Theorem for oriented graphs and consequences on digraph redicolouring

Picasarri-Arrieta¹

2023

Preprint

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Let D = (V, A) be a digraph. We define ∆max(D) as the maximum of {max(dIt is known that the dichromatic number of D is at most ∆min(D) + 1. In this work, we prove that every digraph D which has dichromatic number exactly ∆min(D) + 1 must contain the directed join of ← → Kr and ← → Ks for some r, s such that r + s = ∆min(D) + 1. In particular, every oriented graph G with ∆min( G) ≥ 2 has dichromatic number at most ∆min( G).Let G be an oriented graph of order n such that ∆min( G) ≤ 1. Given two 2-dicolourings of G, we show that we can transform one into the other in at most n steps, by recolouring one vertex at each step while maintaining a dicolouring at any step. Furthermore, we prove that, for every oriented graph G on n vertices, the distance between two k-dicolourings is at most 2∆min( G)n when k ≥ ∆min( G) + 1.We then extend a theorem of Feghali to digraphs. We prove that, for every digraph D with ∆max(D) = ∆ ≥ 3 and every k ≥ ∆ + 1, the k-dicolouring graph of D consists of isolated vertices and at most one further component that has diameter at most c∆n 2 , where c∆ = O(∆ 2 ) is a constant depending only on ∆.

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Four proofs of the directed Brooks' Theorem

Abstract: We give four new proofs of the directed version of Brook's Theorem and an NP-completeness result.

Cited by 2 publications

References 22 publications

Partitioning into degenerate graphs in linear time

Partitioning into degenerate graphs in linear time

Strengthening the Directed Brooks' Theorem for oriented graphs and consequences on digraph redicolouring

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