Four proofs of the directed Brooks' Theorem

Aboulker, Pierre; Aubian, Guillaume

doi:10.1016/j.disc.2022.113193

Cited by 4 publications

(5 citation statements)

References 23 publications

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“…Similarly, it is well known (see Lemma 12(1)) that every vertex in a k-dicritical digraph has degree at least 2(k −1) and hence a k-dicritical digraph has at least (k −1)|V (G)| arcs. Brooks' theorem was generalised in [Moh10] (see also [AA22]) to digraphs, and implies a simple characterisation of the k-dicritical digraphs G with exactly (k − 1)|V (G)| arcs. For G a digraph, let ∆ max (G) be the maximum over the vertices of G of the maximum of their in-degree and their out-degree.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Various Bounds on the Minimum Number of Arcs in a $k$-Dicritical Digraph

Aboulker,

Vermande

2024

Electron. J. Combin.

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The dichromatic number $\vec{\chi}(G)$ of a digraph $G$ is the least integer $k$ such that $G$ can be partitioned into $k$ acyclic digraphs. A digraph is $k$-dicritical if $\vec{\chi}(G) = k$ and each proper subgraph $H$ of $G$ satisfies $\vec{\chi}(H) \leq k-1$. We prove various bounds on the minimum number of arcs in a $k$-dicritical digraph, a structural result on $k$-dicritical digraphs and a result on list-dicolouring. We characterise $3$-dicritical digraphs $G$ with $(k-1)|V(G)| + 1$ arcs. For $k \geq 4$, we characterise $k$-dicritical digraphs $G$ on at least $k+1$ vertices and with $(k-1)|V(G)| + k-3$ arcs, generalising a result of Dirac. We prove that, for $k \geq 5$, every $k$-dicritical digraph $G$ has at least $(k-\frac 1 2 - \frac 1 {k-1}) |V(G)| - k(\frac 1 2 - \frac 1 {k-1})$ arcs, which is the best known lower bound. We prove that the number of connected components induced by the vertices of degree $2(k-1)$ of a $k$-dicritical digraph is at most the number of connected components in the rest of the digraph, generalising a result of Stiebitz. Finally, we generalise a Theorem of Thomassen on list-chromatic number of undirected graphs to list-dichromatic number of digraphs.

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Section: Introduction and Resultsmentioning

confidence: 99%

Various Bounds on the Minimum Number of Arcs in a $k$-Dicritical Digraph

Aboulker,

Vermande

2024

Electron. J. Combin.

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“…The directed version of Brooks' theorem was first proved by Harutyunyan and Mohar in [13]. Aboulker and Aubian gave four new proofs of the following theorem in [1]. .…”

Section: Digraph Colouringmentioning

confidence: 99%

“…. Unfortunately, Aboulker and Aubian [1] proved that, given a digraph D, deciding whether D is D Δ ( ) min -dicolourable is NP-complete. Thus, unless P = NP, we cannot expect an easy characterization of digraphs satisfying…”

Section: Digraph Colouringmentioning

confidence: 99%

“…

k

‐ Dicolourability is NP‐complete for every fixed

k\ge 2

[3]. It remains NP‐complete when we restrict to digraphs

D

with

{{\rm{\Delta }}}_{\min }(D)=k

[1].…”

Section: Proofs Of Theorems 6 Andmentioning

confidence: 99%

“…Hence, one can wonder if Brooks' theorem can be extended to digraphs using

{{\rm{\Delta }}}_{\min }(D)

instead of

{{\rm{\Delta }}}_{\max }(D)

. Unfortunately, Aboulker and Aubian [1] proved that, given a digraph

D

, deciding whether

D

{{\rm{\Delta }}}_{\min }(D)

‐dicolourable is NP‐complete. Thus, unless P = NP, we cannot expect an easy characterization of digraphs satisfying

\overrightarrow{\chi }(D)={{\rm{\Delta }}}_{\min }(D)+1

.…”

Section: Introductionmentioning

confidence: 99%

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Strengthening the directed Brooks' theorem for oriented graphs and consequences on digraph redicolouring

Picasarri‐Arrieta

2023

Journal of Graph Theory

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Let be a digraph. We define as the maximum of and as the maximum of . It is known that the dichromatic number of is at most . In this work, we prove that every digraph which has dichromatic number exactly must contain the directed join of and for some such that , except if in which case must contain a digon. In particular, every oriented graph with has dichromatic number at most . Let be an oriented graph of order such that . Given two 2‐dicolourings of , we show that we can transform one into the other in at most steps, by recolouring exactly one vertex at each step while maintaining a dicolouring at any step. Furthermore, we prove that, for every oriented graph on vertices, the distance between two ‐dicolourings is at most when . We then extend a theorem of Feghali, Johnson and Paulusma to digraphs. We prove that, for every digraph with and every , the ‐dicolouring graph of consists of isolated vertices and at most one further component that has diameter at most , where is a constant depending only on .

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