Coloring graphs with no induced five‐vertex path or gem

Chudnovsky, Maria; Karthick, T.; Maceli, Peter; Maffray, Frédéric

doi:10.1002/jgt.22572

Cited by 25 publications

(24 citation statements)

References 16 publications

(28 reference statements)

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“…Let T = N S 1 (s 1 ) ∩ N S 1 (s 2 ), let T 1 = N S 1 (s 1 ) \ T , and let T 2 = N S 1 (s 2 ) \ T . If T i is not a clique for some i ∈ {1, 2}, let t i,1 and t i,2 be two non-adjacent vertices of T i , then {s 3−i , t i,1 , t i,2 } is an independent set in S 1 , which leads to a contradiction to (8). Thus both T 1 and T 2 are cliques.…”

Section: Cliques As They Are Subsets Of S and Thusmentioning

confidence: 97%

“…Note that S 1 = N (v 1 ). By (8), we have that no vertex of N (v 1 )\{s 1 , s 2 } is anticomplete to {s 1 , s 2 }. Let T = N S 1 (s 1 ) ∩ N S 1 (s 2 ), let T 1 = N S 1 (s 1 ) \ T , and let T 2 = N S 1 (s 2 ) \ T .…”

Section: Cliques As They Are Subsets Of S and Thusmentioning

confidence: 99%

“…Dong and Xu [10] proved that (P 5 , F )-free graphs are perfectly divisible, where F is either a cochair or a cricket. Chudnovsky et al [8] proved that χ(G) ≤ ⌈ 5ω (G) 4 ⌉ if G is (P 5 , gem)-free, which improves the results of [4] and [9]. Char and Karthick [5] showed that if G is (P…”

Section: Introductionmentioning

confidence: 96%

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On the chromatic number of some $P_5$-free graphs

Wang¹,

Xu²,

Xu³

2022

Preprint

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Let G be a graph. We say that G is perfectly divisible if for each induced subgraph H of G, V (H) can be partitioned into A and B such that H[A] is perfect and ω(H[B]) < ω(H). We use P t and C t to denote a path and a cycle on t vertices, respectively. For two disjoint graphs F 1 and F 2 , we use F 1 ∪ F 2 to denote the graph with vertex set V (F 1 ) ∪ V (F 2 ) and edge set E(F 1 ) ∪ E(F 2 ), and use F 1 + F 2 to denote the graph with vertex setIn this paper, we prove that (i) (P 5 , C 5 , K 2,3 )-free graphs are perfectly divisible, (ii)free, and (iv) χ(G) ≤ 3ω(G)+11 if G is (P 5 , K 1 +(K 1 ∪K 3 ))-free.

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Section: Cliques As They Are Subsets Of S and Thusmentioning

confidence: 97%

Section: Cliques As They Are Subsets Of S and Thusmentioning

confidence: 99%

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On the chromatic number of some $P_5$-free graphs

Wang¹,

Xu²,

Xu³

2022

Preprint

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“…Fouquet et al [9] proved that there are infinitely many (

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

)‐free graphs

G

with

χ (G) \geq ω {MathClass-open(G MathClass-close)}^{μ}

, where

μ = \log_{2} 5 - 1

, and that every (

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

)‐free graph

G

satisfies

χ MathClass-open(G MathClass-close) \leq true(\frac{0}{ω (G) + 1} true)

. Very recently, Chudnovsky et al [6] showed that every (

P_{5}

K_{1} + P_{4}

)‐free graph

G

satisfies

χ MathClass-open(G MathClass-close) \leq true⌈ \frac{5 ω (G)}{4} true⌉

. We refer to a recent comprehensive survey of Schiermeyer and Randerath [18] for more results.…”

Section: Introductionmentioning

confidence: 99%

On graphs with no induced five‐vertex path or paraglider

Huang

Karthick

2021

Journal of Graph Theory

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Given two graphs H 1 and H 2, a graph is ( H 1 , H 2 )‐free if it contains no induced subgraph isomorphic to H 1 or H 2. For a positive integer t, P t is the chordless path on t vertices. A paraglider is the graph that consists of a chorless cycle C 4 plus a vertex adjacent to three vertices of the C 4. In this paper, we study the structure of ( P 5, paraglider)‐free graphs, and show that every such graph G satisfies χ MathClass-open( G MathClass-close) ≤ true⌈ 3 2 ω MathClass-open( G MathClass-close) true⌉, where χ ( G ) and ω ( G ) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the Clebsch graph on 16 vertices. More strongly, we completely characterize all the ( P 5, paraglider)‐free graphs G that satisfies χ ( G ) > 3 2 ω ( G ). We also construct an infinite family of ( P 5, paraglider)‐free graphs such that every graph G in the family has χ MathClass-open( G MathClass-close) = true⌈ 3 2 ω MathClass-open( G MathClass-close) true⌉ − 1. This shows that our upper bound is optimal up to an additive constant and that there is no ( 3 2 − ϵ )‐approximation algorithm for the chromatic number of ( P 5, paraglider)‐free graphs for any ϵ > 0.

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“…The graph P 4 ∨ K 1 is called a gem, and a co-gem is the complement of a gem. Cameron, Huang and Merkel [3] showed that χ(G) ≤ ⌊ 3 2 ω(G)⌋ for every (P 5 , gem)-free graph G. Chudnovsky, Karthick, Maceli, and Maffray [6] improved the χ-binding function of (P 5 , gem)-free graphs to ⌈ 5 4 ω(G)⌉. Karthick and Maffray [16] showed that χ(G) ≤ ⌈ 5 4 ω(G)⌉ for every (gem, co-gem)-free graph G, and in [18] they showed that every (P 5 , diamond)-free graph G satisfies χ(G) ≤ ω(G) + 1.…”

Section: Introductionmentioning

confidence: 99%

Colouring graphs with no induced six-vertex path or diamond

Goedgebeur¹,

Huang²,

Ju³

et al. 2021

Preprint

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The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P 6 , diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P 6 , diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We do this by reducing the problem to imperfect (P 6 , diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical (P 6 , diamond, K 6 )-free graph. Together with the Lovász theta function, this gives a polynomial time algorithm to compute the chromatic number of (P 6 , diamond)-free graphs.

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Coloring graphs with no induced five‐vertex path or gem

Abstract: For a graph G, let χ G () and ω G (), respectively, denote the chromatic number and clique number of G. We give an explicit structural description of (P 5 , gem)-free graphs, and show that every such graph G satisfies χ G () ω G 5 () 4

Cited by 25 publications

References 16 publications

On the chromatic number of some $P_5$-free graphs

On the chromatic number of some $P_5$-free graphs

On graphs with no induced five‐vertex path or paraglider

Colouring graphs with no induced six-vertex path or diamond

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