Colouring Vertices of Triangle-Free Graphs

Dabrowski, Konrad K.; Raman, Rajiv; Ries, Bernard

doi:10.1007/978-3-642-16926-7_18

Cited by 10 publications

(8 citation statements)

References 24 publications

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“…Some initial results have been obtained by Král' et al [15], Schindl [22] and a number of authors studying the H-free graphs, in which one of the two graphs in H is the triangle [5,7,13,19]. Another extension is to classify the computational complexity of k-Coloring and other variants of coloring for H-free graphs where k is a fixed integer and H is a fixed graph.…”

Section: Theorem 1 ([15])mentioning

confidence: 99%

List Coloring in the Absence of a Linear Forest

Couturier¹,

Golovach

Kratsch³

et al. 2013

Algorithmica

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The nal publication is available at Springer via http://dx.doi.org/10.1007/s00453-013-9777-0.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1, . . . , k}.Let Pn denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that List k-Coloring can be solved in polynomial time for graphs with no induced rP1 + P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1 + P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H.

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Section: Theorem 1 ([15])mentioning

confidence: 99%

List Coloring in the Absence of a Linear Forest

Couturier¹,

Golovach

Kratsch³

et al. 2013

Algorithmica

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“…Theorem 1 can be extended by considering the computational complexity of Coloring for H-free graphs where H is a family of two (or more) graphs. Some initial results have been obtained by Král' et al [15], Schindl [22] and a number of authors studying the H-free graphs, in which one of the two graphs in H is the triangle [5,7,13,19]. Another extension is to classify the computational complexity of k-Coloring and other variants of coloring for H-free graphs where k is a fixed integer and H is a fixed graph.…”

Section: Theorem 1 ([15]mentioning

confidence: 99%

List Coloring in the Absence of a Linear Forest

Golovach

Paulusma

2011

Graph-Theoretic Concepts in Computer Science

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The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1,. .. , k}. Let Pn denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that List k-Coloring can be solved in polynomial time for graphs with no induced rP1 + P5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P5. Our result is tight; we prove that for any graph H that is a supergraph of P1 + P5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H.

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“…In this paper we consider graph classes defined by a forbidden induced subgraph. In contrast to previous papers on this topic [3][4][5][6]8,10,12,[15][16][17][18][19]21,23,27], we focus on the parameterized complexity. Before we summarize these results and explain our new results, we first state the necessary terminology and notations.…”

Section: Introductionmentioning

confidence: 98%

On the parameterized complexity of coloring graphs in the absence of a linear forest

Couturier¹,

Golovach²,

Kratsch³

et al. 2012

Journal of Discrete Algorithms

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List k-Coloring H-free Linear forest Fixed-parameter tractableThe k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1, . . . ,k}. Let P n denote the path on n vertices, and G + H and r H the disjoint union of two graphs G and H and r copies of H, respectively. We show that List k-Coloring is fixedparameter tractable on graphs with no induced r P 1 + P 2 when parameterized by k + r, and that for any fixed integer r, the problem k-Coloring restricted to such graphs allows a polynomial kernel when parameterized by k. Finally, we show that List k-Coloring is fixed-parameter tractable on graphs with no induced P 1 + P 3 when parameterized by k.

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Colouring Vertices of Triangle-Free Graphs

Cited by 10 publications

References 24 publications

List Coloring in the Absence of a Linear Forest

List Coloring in the Absence of a Linear Forest

List Coloring in the Absence of a Linear Forest

On the parameterized complexity of coloring graphs in the absence of a linear forest

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