Three-Coloring and List Three-Coloring of Graphs Without Induced Paths on Seven Vertices

Bonomo, Flavia; Chudnovsky, Mariana; Maceli, Peter; Schaudt, Oliver; Stein, Maya; Zhong, Mingxian

doi:10.1007/s00493-017-3553-8

Cited by 80 publications

(143 citation statements)

References 24 publications

(19 reference statements)

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“…Their proof can be used to show that List 3 ‐Colouring is polynomial‐time solvable on

(P_{2} + P_{4})

‐free graphs. Recently, Bonomo, Chudnovsky, Maceli, Schaudt, Stein and Zhong gave an

O (| V {false|}^{23} |)

time algorithm for List 3‐Colouring on P 7 ‐free graphs (thereby solving Problem 17 in and Problem 56 in ). The same authors showed that 3‐Colouring can be solved in

O (| V {false|}^{7} |)

time on

(C_{3}, P_{7})

‐free graphs. …”

Section: Results and Open Problems For H‐free Graphsmentioning

confidence: 99%

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

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

130

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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“…Their proof can be used to show that List 3 ‐Colouring is polynomial‐time solvable on

(P_{2} + P_{4})

‐free graphs. Recently, Bonomo, Chudnovsky, Maceli, Schaudt, Stein and Zhong gave an

O (| V {false|}^{23} |)

time algorithm for List 3‐Colouring on P 7 ‐free graphs (thereby solving Problem 17 in and Problem 56 in ). The same authors showed that 3‐Colouring can be solved in

O (| V {false|}^{7} |)

time on

(C_{3}, P_{7})

‐free graphs. …”

Section: Results and Open Problems For H‐free Graphsmentioning

confidence: 99%

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

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

130

160

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show abstract

“…This proves (5.2). 2 is an induced P 6 in G, a contradiction. This shows that y 1 ∈ Z k and k ∈ {i, j}, or that y 2 ∈ Z and ∈ {i, j}.…”

Section: Contradiction This Proves (422) This Proves (4)mentioning

confidence: 77%

“…Thus we may assume that x is nonadjacent to v 2 , and therefore we obtain outcome (2.1) for i = 0. This proves (2 …”

Section: Lemmamentioning

confidence: 81%

“…For t = 5, a polynomial-time algorithm [15] exists for every fixed k , but the exponent of the polynomial heavily depends on k. For t ≥ 6, because of the above hardness results, there are only two interesting remaining cases: t = 6 and k = 4, and t ≥ 6 and k = 3. The latter has been established for t = 6 [4,24] and very recently also for t = 7 [2], while the situation when t ≥ 8 remains wide open. [14] NPc NPc *NPc for t = 22 [18].…”

Section: Introductionmentioning

confidence: 94%

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

et al. 2016

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“…As coloring is polynomial‐time solvable for

(P_{5}, \overset{P_{5}}{true})

‐free graphs , it would be interesting to solve the following two open problems (see also Table ): is there an integer

k

such that

k

‐ coloring for

(P_{6}, \overset{P_{6}}{true})

‐free graphs is

sans-serif𝖭𝖯

‐complete? is there an integer

k

such that

k

‐ coloring for

(P_{7}, \overset{P_{7}}{true})

‐free graphs is

sans-serif𝖭𝖯

‐complete? As

4

‐ coloring is polynomial‐time solvable for

P_{6}

‐free graphs ,

k

must be at least

5

in an affirmative answer for the first question. As

3

‐ coloring is polynomial‐time solvable for

P_{7}

‐free graphs ,

k

must be at least

4

in an affirmative answer for the second question. We refer to the end of Section for some further remarks on these two open problems.…”

Section: Introductionmentioning

confidence: 99%

Hereditary graph classes: When the complexities of coloring and clique cover coincide

Blanché

Dabrowski

Johnson

et al. 2018

Journal of Graph Theory

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A graph is ( H 1 , H 2 )‐free for a pair of graphs H 1 , H 2 if it contains no induced subgraph isomorphic to H 1 or H 2. In 2001, Král et al initiated a study into the complexity of coloring for ( H 1 , H 2 )‐free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those ( H 1 , H 2 )‐free graphs where H 2 is H true¯ 1, the complement of H 1. As these classes are closed under complementation, the computational complexities of coloring and clique cover coincide. By combining new and known results, we are able to classify the complexity of coloring and clique cover for ( H , trueH ¯ )‐free graphs for all cases except when H = s P 1 + P 3 for s ≥ 3 or H = s P 1 + P 4 for s ≥ 2. We also classify the complexity of coloring on graph classes characterized by forbidding a finite number of self‐complementary induced subgraphs, and we initiate a study of k‐ coloring for ( P r , P true¯ r )‐free graphs.

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Three-Coloring and List Three-Coloring of Graphs Without Induced Paths on Seven Vertices

Abstract: We present an algorithm to 3-colour a graph G without triangles or induced paths on seven vertices in O(|V (G)| 7 ) time. In fact, our algorithm solves the list 3-colouring problem, where each vertex is assigned a subset of {1, 2, 3} as its admissible colours.

Cited by 80 publications

References 24 publications

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

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

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

Hereditary graph classes: When the complexities of coloring and clique cover coincide

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Three-Coloring and List Three-Coloring of Graphs Without Induced Paths on Seven Vertices

Abstract: We present an algorithm to 3-colour a graph G without triangles or induced paths on seven vertices in O(|V (G)| 7 ) time. In fact, our algorithm solves the list 3-colouring problem, where each vertex is assigned a subset of {1, 2, 3} as its admissible colours.

Cited by 80 publications

References 24 publications

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

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

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

Hereditary graph classes: When the complexities of coloring and clique cover coincide

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