4‐Coloring
            <i>P</i>
            <sub>6</sub>
            ‐Free Graphs with No Induced 5‐Cycles

Chudnovsky, Maria; Maceli, Peter; Stacho, Juraj; Zhong, Mingxian

doi:10.1002/jgt.22025

Cited by 10 publications

(8 citation statements)

References 33 publications

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“…They might be a first step in the attempt to settle the case of P t -free graphs. This turned out to be the case for 3-coloring of P 7 -free graphs (as can be seen from preprints [2,7,8] leading to [1]) and 4-coloring of P 6 -free graphs [6].…”

Section: Introductionmentioning

confidence: 88%

On 3-Coloring of $(2P_4,C_5)$-Free Graphs

Jelínek,

Klimošová,

Masařík

et al. 2020

Preprint

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Section: Introductionmentioning

confidence: 88%

On 3-Coloring of $(2P_4,C_5)$-Free Graphs

Jelínek,

Klimošová,

Masařík

et al. 2020

Preprint

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“…Suppose H is a linear forest. We first recall that List 3-Colouring is polynomial-time solvable for P 7 -free graphs [8] and thus for (rP 1 + P 7 )-free graphs for every integer r ≥ 0 [5,15]. Now suppose that H is not an induced subgraph of rP 1 + P 7 for any r ≥ 0.…”

Section: The Proof Of Corollarymentioning

confidence: 99%

Colouring (Pr + Ps)-Free Graphs

et al. 2020

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The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) ⊆ {1, … , k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph P r + P s is the disjoint union of the r-vertex path P r and the s-vertex path P s. We prove that List 3-Colouring is polynomial-time solvable for (P 2 + P 5)-free graphs and for (P 3 + P 4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices.

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“…Hence, it remains to consider the case where H is a linear forest (a disjoint union of paths). The situation is far from settled yet, although many partial results are known [3,4,5,7,8,9,10,11,16,20,22,26,30,32,35]. Particularly, the case where H is the t-vertex path P t has been well studied.…”

Section: List K-colouringmentioning

confidence: 99%

Colouring $(P_r+P_s)$-Free Graphs

Klimošová,

Malík,

Masařík

et al. 2018

Preprint

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The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) ⊆ {1, . . . , k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph Pr + Ps is the disjoint union of the r-vertex path Pr and the s-vertex path Ps. We prove that List 3-Colouring is polynomial-time solvable for (P2 + P5)-free graphs and for (P3 + P4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P3 + P5)-free graphs.

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

Abstract: Abstract:We show that the 4-coloring problem can be solved in polynomial time for graphs with no induced 5-cycle C 5 and no induced 6-vertex path P 6 C 2016 Wiley Periodicals, Inc. J. Graph Theory 00: 2016

Cited by 10 publications

References 33 publications

On 3-Coloring of $(2P_4,C_5)$-Free Graphs

On 3-Coloring of $(2P_4,C_5)$-Free Graphs

Colouring (Pr + Ps)-Free Graphs

Colouring $(P_r+P_s)$-Free Graphs

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

Abstract: Abstract:We show that the 4-coloring problem can be solved in polynomial time for graphs with no induced 5-cycle C 5 and no induced 6-vertex path P 6 C 2016 Wiley Periodicals, Inc. J. Graph Theory 00: 2016

Cited by 10 publications

References 33 publications

On 3-Coloring of $(2P_4,C_5)$-Free Graphs

On 3-Coloring of $(2P_4,C_5)$-Free Graphs

Colouring (Pr + Ps)-Free Graphs

Colouring $(P_r+P_s)$-Free Graphs

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