Dichromatic Number and Fractional Chromatic Number

Mohar, Bojan; Wu, Hehui

doi:10.1017/fms.2016.28

Cited by 20 publications

(17 citation statements)

References 11 publications

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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: 91%

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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Section: Dichromatic Numbermentioning

confidence: 91%

Parameterized Algorithms for Directed Modular Width

Steiner

Wiederrecht

2020

Lecture Notes in Computer Science

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“…We will use the following lemmas from our earlier work [19]. For the sake of completeness we include the proofs here as well.…”

Section: Fractional Weight and Principal Vertex-setsmentioning

confidence: 99%

“…They are based on two concepts, that of a principal subset of vertices and that of a sparse subset. These two notions were used previously in our fractional versions of the Erdős‐Neumann‐Lara conjecture (see [19]) and the Erdős‐Hajnal conjecture from [12] for triangle‐free subgraphs (see [20]) that every graph with large chromatic number contains a triangle‐free subgraph whose chromatic number is still large.…”

Section: Fractional Weight and Principal Vertex‐setsmentioning

confidence: 99%

Fractional chromatic number of a random subgraph

Mohar

2020

Journal of Graph Theory

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It is well known [6] that a random subgraph of the complete graph K n has chromatic number Θ(n/ log n) w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any n-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least n, the fractional chromatic number of its random subgraph is at least n/(8 log 2 (4n)) with probability more than 1 − 1 2n . This gives the affirmative answer for a strengthening of Bukh's question for the fractional chromatic number.

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“…The next claim from [10] about the total weight of an s-sparse set will be essential for us. We include the proof for completeness.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Triangle-free subgraphs with large fractional chromatic number

Mohar

2015

Electronic Notes in Discrete Mathematics

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It is well known that for any integers k and g, there is a graph with chromatic number at least k and girth at least g. In 1960's, Erdős and Hajnal conjectured that for any k and g, there exists a number h(k, g), such that every graph with chromatic number at least h(k, g) contains a subgraph with chromatic number at least k and girth at least g. In 1977, Rödl proved the case for g = 4 and arbitrary k. We prove the fractional chromatic number version of Rödl's result.

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Dichromatic Number and Fractional Chromatic Number

Cited by 20 publications

References 11 publications

Parameterized Algorithms for Directed Modular Width

Parameterized Algorithms for Directed Modular Width

Fractional chromatic number of a random subgraph

Triangle-free subgraphs with large fractional chromatic number

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