List coloring digraphs

Bensmail, Julien; Harutyunyan, Ararat; Le, Ngoc Khang

doi:10.1002/jgt.22170

Cited by 24 publications

(31 citation statements)

References 21 publications

(55 reference statements)

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“…Moreover, Bensmail et al managed to extend the above theorem to list‐colorings of digon‐free digraphs.…”

Section: Discussionmentioning

confidence: 99%

“…In , a simple transformation is used to obtain the directed version of Ohba's Conjecture from the undirected case.…”

Section: Discussionmentioning

confidence: 99%

“…In an earlier version of the present paper, the authors conjectured that it should be possible to transfer the above theorem to DP‐colorings of directed graphs. Since then, we have managed to find a way to achieve this by combining the ideas from with a similar technique to the one used in the proof of Theorem .…”

Section: Discussionmentioning

confidence: 99%

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On DP‐coloring of digraphs

Bang‐Jensen

Bellitto

Schweser

et al. 2019

Journal of Graph Theory

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DP-coloring is a relatively new coloring concept by Dvořák and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph G with a list-assignment L to finding an independent transversal in an auxiliary graph with vertex setIn this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.

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“…Moreover, Bensmail et al managed to extend the above theorem to list‐colorings of digon‐free digraphs.…”

Section: Discussionmentioning

confidence: 99%

“…In , a simple transformation is used to obtain the directed version of Ohba's Conjecture from the undirected case.…”

Section: Discussionmentioning

confidence: 99%

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On DP‐coloring of digraphs

Bang‐Jensen

Bellitto

Schweser

et al. 2019

Journal of Graph Theory

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“…It has become apparent that the dichromatic number acts as a natural directed counterpart of the chromatic number of an undirected graph. Numerous recent results (see [AH15], [MW16], [ACH + 16], [LM17], [BHKL18], [HLTW19], [MSW19]) support this claim. Formally, we consider the following problem.…”

Section: Dichromatic Numbermentioning

confidence: 90%

Parameterized Algorithms for Directed Modular Width

Steiner

Wiederrecht

2020

Lecture Notes in Computer Science

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Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investigate directed modular width as a parameter, which is closer to the concept of clique-width. We investigate applications of modular decompositions of directed graphs to a wide range of algorithmic problems and derive FPT-algorithms for several well-known digraph-specific NP-hard problems, namely minimum (weight) directed feedback vertex set, minimum (weight) directed dominating set, digraph colouring, directed Hamiltonian path/cycle, partitioning into paths, (capacitated) vertex-disjoint directed paths, and the directed subgraph homeomorphism problem. The latter yields a polynomial-time algorithm for detecting topological minors in digraphs of bounded directed modular width. Finally we illustrate that also other structural digraph parameters, such as directed pathwidth and the cycle-rank can be computed efficiently using directed modular width as a parameter.

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“…A surge of recent results on the subject suggests that this concept is accepted as the natural extension of the chromatic number of graphs to digraphs by most authors. For example see: [3,7,15,20,22,25,26,30]. Note that the problem of deciding if a digraph has dichromatic number at most 2 is NP-complete (see [8]), even when restricted to tournaments (see [9]), which is in contrast to the undirected case, where 2-colorability is polynomial time to decide.…”

Section: Introductionmentioning

confidence: 99%

Extension of Gyárfás-Sumner Conjecture to Digraphs

Aboulker¹,

Charbit

Naserasr³

2021

Electron. J. Combin.

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The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph. Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

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List coloring digraphs

Cited by 24 publications

References 21 publications

On DP‐coloring of digraphs

On DP‐coloring of digraphs

Parameterized Algorithms for Directed Modular Width

Extension of Gyárfás-Sumner Conjecture to Digraphs

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