Induced trees in graphs of large chromatic number

Scott, Alex

doi:10.1002/(sici)1097-0118(199704)24:4<297::aid-jgt2>3.0.co;2-j

Cited by 81 publications

(88 citation statements)

References 12 publications

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“…The main result of this paper is the following theorem that takes a significant step in this direction. Our proof uses all the techniques of [5], techniques from [7] and [4], as well as several new ideas. Theorem 1.6.…”

Section: Theorem 13 (Kierstead and Penricementioning

confidence: 99%

“…In the remainder of this section we introduce our notation. In section 2 we prove a local coloring result based on the work of Scott [7]. In section 3 we present several general coloring techniques that will be used later.…”

Section: Theorem 13 (Kierstead and Penricementioning

confidence: 99%

“…In section 5 we describe a partitioning of the vertices of a graph based on 1-and 2-neighborhoods of templates, and in section 6 we use this partition to color the vertices of the graph. The novelty of this work involves adapting the template technique of Kierstead and Penrice [5] to take advantage of the new local colorings made available by Scott's ideas [7]. This adaptation is not straightforward.…”

Section: Theorem 13 (Kierstead and Penricementioning

confidence: 99%

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Radius Three Trees in Graphs with Large Chromatic Number

Kierstead

Zhu

2004

SIAM J. Discrete Math.

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A class Γ of graphs is χ-bounded if there exists a function f such that χ (G) ≤ f (ω (G)) for all graphs G ∈ Γ, where χ denotes chromatic number and ω denotes clique number. Gyárfás and Sumner independently conjectured that, for any tree T , the class Forb (T ), consisting of graphs that do not contain T as an induced subgraph, is χ-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.

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Section: Theorem 13 (Kierstead and Penricementioning

confidence: 99%

Section: Theorem 13 (Kierstead and Penricementioning

confidence: 99%

Section: Theorem 13 (Kierstead and Penricementioning

confidence: 99%

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Radius Three Trees in Graphs with Large Chromatic Number

Kierstead

Zhu

2004

SIAM J. Discrete Math.

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“…A class of graphs is χ -bounded if there exists a χ -bounding function that holds for all graphs of the class. Scott [15] conjectured that for any graph H , the class of those graphs that do not contain any subdivision of H as an induced subgraph is χ -bounded. This conjectured was disproved by Pawlik et al [13].…”

Section: Theorem 12 Let G Be An {Isk4 Wheel}-free Graph Then Eithermentioning

confidence: 99%

On graphs with no induced subdivision of K4

Lévêque

Maffray

Trotignon

2012

Journal of Combinatorial Theory, Series B

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We prove a decomposition theorem for graphs that do not contain a subdivision of K 4 as an induced subgraph where K 4 is the complete graph on four vertices. We obtain also a structure theorem for the class C of graphs that contain neither a subdivision of K 4 nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in C is 3-colorable and entails a polynomial-time recognition algorithm for membership in C. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless.

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“…Indeed, a complete bipartite graph has no clique of size 3, but contains only stars as induced subtrees. This conjecture is widely open, although some partial results were obtained in [10][11][12]18].…”

Section: Theorem 2 Let R 4 and Let G Be A Connected Graph On N Vertmentioning

confidence: 99%

Large induced trees in Kr-free graphs

Fox¹,

Loh²,

Sudakov³

2009

Journal of Combinatorial Theory, Series B

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For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K r on r vertices. This problem was posed twenty years ago by Erdős, Saks, and Sós. Substantially improving earlier results of various researchers, we prove that every connected triangle-free graph on n vertices contains an induced tree of order √ n. When r 4, we also show that t(G) log n 4 log r for every connected K r -free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matoušek and Šámal.

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Induced trees in graphs of large chromatic number

Cited by 81 publications

References 12 publications

Radius Three Trees in Graphs with Large Chromatic Number

Radius Three Trees in Graphs with Large Chromatic Number

On graphs with no induced subdivision of K4

Large induced trees in Kr-free graphs

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