Updating the complexity status of coloring graphs without a fixed induced linear forest

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.

Section: Theorem 1 ([15])mentioning

confidence: 98%

“…It is known that 4-Coloring is NP-complete for P 8 -free graphs [4] and that 6-Coloring is NP-complete for P 7 -free graphs [3]. On the contrary, Randerath and Schiermeyer [20] showed that 3-Coloring can be solved in polynomial time for P 6 -free graphs.…”

Section: Theorem 1 ([15])mentioning

confidence: 99%

List Coloring in the Absence of a Linear Forest

Couturier¹,

Kratsch³

et al. 2013

Algorithmica

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“…This shows Case (ii):5, whereas we obtain Case (ii):4 by using the same arguments together with a result of Král' et al [24], who showed that for any fixed graph H 2 , Coloring is polynomialtime solvable on (C 3 , H 2 )-free graphs if and only if it is so for (C + 3 , H 2 )-free graphs. Case (ii):3 is proven by combining the latter result with corresponding results of Dabrowski, Lozin, Raman and Ries [11] for (C 3 , H 2 )-free graphs that are obtained by combining a number of new results with known results for H 2 = K 1,4 [24], H 2 = S 1,2,2 [32], H 2 = P 2 + P 4 [6], H 2 = 2P 3 [7], H 2 = P 6 [3], H 2 is the cross [33] (the graph obtained from K 1,4 by making a new vertex adjacent to one of its leafs) and H 2 is the 'H'-graph [32] (the graph obtained from K 1,3 by making two new non-adjacent vertices adjacent to the same leaf). Finally, Case (ii):8 has been shown by Dabrowski, Golovach and Paulusma [10].…”

Section: Related Workmentioning

confidence: 83%

List coloring in the absence of two subgraphs

Discrete Applied Mathematics

Paulusma

2014

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Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Discrete applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication. A de nitive version was subsequently published in Discrete applied mathematics, 166, 2014, 10.1016/j.dam.2013.10.010 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. The List Coloring problem is that of testing whether a given graph G = (V, E) has a coloring c that respects a given list assignment L, i.e., whether G has a mapping c :If a graph G has no induced subgraph isomorphic to some graph of a pair {H1, H2}, then G is called (H1, H2)-free. We completely characterize the complexity of List Coloring for (H1, H2)-free graphs.

“…Then we may assume without loss of generality that d G (u) ≥ k for all u ∈ V (G), as otherwise we perform the following well-known procedure (see e.g. [5]). We repeatedly delete a vertex with degree at most k − 1 from G until no such vertex remains.…”

Section: Case 1 H Has No Dominating Vertexmentioning

confidence: 99%

“…The latter follows from the fact that there exist combinations of integers k and linear forests H for which the k-Coloring problem for H-free graphs is NP-complete; for example, it is known that 4-Coloring is NP-complete for P 8 -free graphs [5] and that 6-Coloring is NP-complete for P 7 -free graphs [4]. Only very few parameterized results for Coloring on H-free graphs are known.…”

Section: Introductionmentioning

confidence: 99%

Choosability on H-free graphs

Information Processing Letters

Heggernes

Hof

et al. 2013

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The 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.