On the chromatic number of (P5, K2,)-free graphs

Brause, Christoph; Doan, Trung Duy; Schiermeyer, Ingo

doi:10.1016/j.endm.2016.10.032

Cited by 10 publications

(13 citation statements)

References 6 publications

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“…Theorem 1.2 improves a result of Brause et al [1] and the upper bound 2ω 2 (G)−ω(G)−3 is sharp in the sense that all (P 5 , K 3 )-free graphs are 3-colorable and there are (P 5 , K 3 )free graphs with chromatic number 3, Theorem 1.3 improves a result of Schiermeyer [21], and Theorem 1.4 improves a result of [26] which states that χ(G)…”

Section: Introductionsupporting

confidence: 83%

“…In this paper, we study a subclasses of P 5 -free graphs, and prove the following theorems, which improve some results of [1,21,26].…”

Section: Introductionmentioning

confidence: 79%

“…Huang and Karthick [18] showed that if G is (P 5 , paraglider)-free, then χ(G) ≤ ⌈ 3ω(G) 2 ⌉. Chudnovsky and Sivaraman [7] showed that χ(G) ≤ 2 ω(G)−1 if G is (P 5 , C 5 )-free, Brause et al [1] proved that χ(G) ≤ d • ω 3 (G) for some constant d if G is (P 5 , K 2,3 )-free, and Schiermeyer [21]…”

Section: Introductionmentioning

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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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: Introductionsupporting

confidence: 83%

“…In this paper, we study a subclasses of P 5 -free graphs, and prove the following theorems, which improve some results of [1,21,26].…”

Section: Introductionmentioning

confidence: 79%

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

Wang¹,

Xu²,

Xu³

2022

Preprint

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“…We further define N (3,1) (C) = ∪ 1≤i≤5 N {i,i+1,i+2} (C), and N (3,2) (C) = ∪ 1≤i≤5 N {i,i+1,i+3} (C). By Lemma 1.1(a), we have…”

Section: Free Graphsmentioning

confidence: 99%

“…Let u be a vertex in Q, and suppose that uv 1 , uv 2 ∈ E(G) by symmetry. Since N (3,1) (C) is anticomplete to N 2 (C) by Lemma 1.1(a), we may choose a vertex, say x, in N {1,2,3,4,5} (C), and let x ′ be a neighbor of x in T . To avoid a…”

Section: And N 2 (C) Can Be Partition Into Two Parts a And B Such Tha...mentioning

confidence: 99%

A tight linear bound to the chromatic number of $(P_5, K_1+(K_1\cup K_3))$-free graphs

Wang¹,

Xu²,

Xu³

2022

Preprint

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Let F 1 and F 2 be two disjoint graphs. The union F 1 ∪ F 2 is a graph with vertex set V (F 1 ) ∪ V (F 2 ) and edge set E(F 1 ) ∪ E(F 2 ), and the joinIn this paper, we present a characterization to (P 5 , K 1 ∪ K 3 )-free graphs, prove that χ(G) ≤ 2ω(G) − 1 if G is (P 5 , K 1 ∪ K 3 )-free. Based on this result, we further prove that χ(G) ≤max{2ω(G), 15} if G is a (P 5 , K 1 + (K 1 ∪ K 3 ))-free graph, and construct an infinite family of (P 5 , K 1 + (K 1 ∪ K 3 ))-free graphs such that every graph G in the family satisfies χ(G) = 2ω(G).

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Chromatic Bounds for the Subclasses of $$pK_2$$-Free Graphs

Prashant

Gokulnath

2021

Algorithms and Discrete Applied Mathematics

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Finding families that admit a linear χ-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of 2K 2 -free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of 2K 2 -free graphs, namely (P 3 ∪ P 2 )-free graphs. We show that the class of {P 3 ∪ P 2 , gem}-free graphs admits f = 2ω as a linear χ-binding function.Furthermore, we give examples to show that the optimal χ-binding function f * ≥ 5ω(G) 4 for the class of {P 3 ∪ P 2 , gem}-free graphs and that the χ-binding function f = 2ω is tight when ω = 2 and 3.

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On the chromatic number of (P5, K2,)-free graphs

Cited by 10 publications

References 6 publications

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

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

A tight linear bound to the chromatic number of $(P_5, K_1+(K_1\cup K_3))$-free graphs

Chromatic Bounds for the Subclasses of $$pK_2$$-Free Graphs

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