Improved complexity results on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-coloring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math>-free graphs

Huang, Shenwei

doi:10.1016/j.ejc.2015.06.005

Cited by 61 publications

(42 citation statements)

References 21 publications

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“…• Make each C-type vertex c ij adjacent to d i and its corresponding literal vertex. We refer to [18] for the proofs of the following two lemmas. To obtain NP-completeness results for (P t , C )-free graphs, we need an additional lemma.…”

Section: The Complexity Of K-coloringmentioning

confidence: 99%

Complexity of coloring graphs without paths and cycles

Hell

Huang

2017

Discrete Applied Mathematics

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Let Pt and C denote a path on t vertices and a cycle on vertices, respectively. In this paper we study the k-coloring problem for (Pt, C )-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of P5-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that k-colorability of P5-free graphs for k ≥ 4 does not. These authors have also shown, aided by a computer search, that 4-colorability of (P5, C5)-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any k, the k-colorability of (P6, C4)-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for k = 3 and k = 4. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring (P6, C4)-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying; in fact they are not efficient in practice, as they depend on multiple use of Ramsey-type results and resulting tree decompositions of very high widths.) To complement these results we show that in most other cases the k-coloring problem for (Pt, C )free graphs is NP-complete. Specifically, for = 5 we show that k-coloring is NP-complete for (Pt, C5)-free graphs when k ≥ 4 and t ≥ 7; for ≥ 6 we show that k-coloring is NP-complete for (Pt, C )-free graphs when k ≥ 5, t ≥ 6; and additionally, for = 7, we show that k-coloring is also NP-complete for (Pt, C7)-free graphs if k = 4 and t ≥ 9. This is the first systematic study of the complexity of the k-coloring problem for (Pt, C )-free graphs. We almost completely classify the complexity for the cases when k ≥ 4, ≥ 4, and identify the last three open cases. no clique cutset, A must be complete to a pair of vertices {x, y} where x ∈ S 3 (v 1 ) and y ∈ S 3 (v 4 ). As G is C 4 -free, A must be a clique and so of size at most k. Moreover, the number of components of S 1 (v 0 ) is at most (k − 2) 2 . Otherwise an induced C 4 would arise by the pigeonhole principle and the fact there are at most (k − 2) 2 pairs of vertices {x, y} with x ∈ S 3 (v 1 ) and y ∈ S 3 (v 4 ). Hence,, is either complete or anti-complete to A, as G is P 6 -free. Let S 3 (v 3 ) and S 3 (v 2 ) be the subsets of S 3 (v 3 ) and S 3 (v 2 ) consisting of all vertices that are complete to A, respectively. Moreover, S 3 (v 3 ) and S 3 (v 2 ) are complete to each other.Our goal is to show that B = ∅ by a similar clique cutset argument. It is not hard to see that every vertex t ∈ T is either complete or anti-complete to B as G is P 6 -free. Let T ⊆ T be the set of those vertices that are complete to A. By the definition of T , any t ∈ T is complete to. Let x ∈ S 5 and t ∈ T be a neighbor of some vertex y ∈ S 3 (v 1 ). Then xytv 3 = C 4 implies that tx ∈ E. Hence, T is complete to S 5 .Next we show that T is a clique. Let t and t be any two vertices in T , and p ∈ B. If t is a n...

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Section: The Complexity Of K-coloringmentioning

confidence: 99%

Complexity of coloring graphs without paths and cycles

Hell

Huang

2017

Discrete Applied Mathematics

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“…However, y ∈ N(w 1 ) ∩ N(w 2 ) and neither 1 nor 2 belongs to L(y), a contradiction. This proves (17).…”

Section: Correctnessmentioning

confidence: 95%

“…NPc [17] NPc [5] NPc [22][27] * k = 5 P [8] P [15] NPc [17] NPc NPc [27] NPc k = 6 P [8] P [15] NPc NPc [4] NPc NPc k ≥ 7 P [8] P [15] NPc NPc NPc NPc k = min P [8] NPc [20] NPc NPc NPc NPc * For t = 12.…”

Section: Introductionmentioning

confidence: 99%

“…It is therefore natural to study the case when H is the t-vertex path P t . This problem has enjoyed a long and exciting history of incremental results [2-4, 14, 15, 17, 18, 22, 24, 25, 27], since the seminal paper [27], and culminating in the most general case proved recently by Shenwei Huang [17] who established that k-COLORING is NP-complete if k ≥ 4 and t ≥ 7, and if k ≥ 5 and t ≥ 6. (See [11] for a survey of these results.)…”

Section: Introductionmentioning

confidence: 99%

“…Namely, 4-COLORING in P 6 -free graphs. Unlike the other outstanding cases, this problem has actually been conjectured to be polynomial-time solvable [17]. Here, we use a more fine-grained approach where, in addition, we exclude a fixed induced -cycle C , and consider the problem in (P t , C )-free graphs.…”

Section: Introductionmentioning

confidence: 99%

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4‐Coloring P ₆ ‐Free Graphs with No Induced 5‐Cycles

et al. 2016

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A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

132

160

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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. For a positive integer k, a k-colouring of a graph G = (V, E) is a mapping c : V → {1, 2, . . . , k} such that c(u) = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a k-colouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem k-COLOURING instead. We survey known results on the computational complexity of COLOURING and k-COLOURING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex. Finally, we also survey results for graph classes defined by some other forbidden pattern.

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Improved complexity results on k-coloring Pt-free graphs

Cited by 61 publications

References 21 publications

Complexity of coloring graphs without paths and cycles

Complexity of coloring graphs without paths and cycles

4‐Coloring P ₆ ‐Free Graphs with No Induced 5‐Cycles

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

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Improved complexity results on k-coloring Pt-free graphs

Cited by 61 publications

References 21 publications

Complexity of coloring graphs without paths and cycles

Complexity of coloring graphs without paths and cycles

4‐Coloring P 6 ‐Free Graphs with No Induced 5‐Cycles

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

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4‐Coloring P ₆ ‐Free Graphs with No Induced 5‐Cycles