Substitution and χ -boundedness

Chudnovsky, Maria; Penev, Irena; Scott, Alex; Trotignon, Nicolas

doi:10.1016/j.jctb.2013.02.004

Cited by 30 publications

(77 citation statements)

References 22 publications

(35 reference statements)

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“…As observed by Penev, for large values of k , a stronger result was implicitly proved in . Theorem If a class of graphs is χ‐bounded by

f false(x false) = x^{A}

, then the closure of the class under substitution is χ‐bounded by

g false(x false) = x^{3 A + 11}

.…”

Section: A Property Closed Under Substitutionsmentioning

confidence: 82%

“…Also gluing along a proper 2‐cutset preserves χ‐boundedness as shown in . Theorem If a class of graphs is χ‐bounded, then its closure under the operation of gluing along a proper 2‐cutset is χ‐bounded.…”

Section: Operations and Properties Preserved By Themmentioning

confidence: 97%

“…This leads to a potential problem: it may happen that an operation O 1 preserves χ‐boundedness, that another operation O 2 also preserves χ‐boundedness, but that the set of operations

{O_{1}, O_{2}}

does not preserves χ‐boundedness. This is explained in , where an actual (but slightly artificial) example of this phenomenon is provided.…”

Section: Operations and Properties Preserved By Themmentioning

confidence: 98%

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χ‐bounds, operations, and chords

Trotignon

Pham

2017

Journal of Graph Theory

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A long unichord in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is long unichord free if it does not contain any long unichord. We prove a structure theorem for long unichord free graph. We give an O(n4m) time algorithm to recognize them. We show that any long unichord free graph G can be colored with at most O(ω3) colors, where ω is the maximum number of pairwise adjacent vertices in G.

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“…As observed by Penev, for large values of k , a stronger result was implicitly proved in . Theorem If a class of graphs is χ‐bounded by

f false(x false) = x^{A}

, then the closure of the class under substitution is χ‐bounded by

g false(x false) = x^{3 A + 11}

.…”

Section: A Property Closed Under Substitutionsmentioning

confidence: 82%

Section: Operations and Properties Preserved By Themmentioning

confidence: 97%

“…This leads to a potential problem: it may happen that an operation O 1 preserves χ‐boundedness, that another operation O 2 also preserves χ‐boundedness, but that the set of operations

{O_{1}, O_{2}}

does not preserves χ‐boundedness. This is explained in , where an actual (but slightly artificial) example of this phenomenon is provided.…”

Section: Operations and Properties Preserved By Themmentioning

confidence: 98%

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χ‐bounds, operations, and chords

Trotignon

Pham

2017

Journal of Graph Theory

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“…This theorem was improved by Chudnovsky, Penev, Scott, and Trotignon [2] who showed that the condition χ(G) > max{c+2k 2 , 2k 2 +k} is sufficient. The proof from [2] relies on an ad hoc induction hypothesis, which roughly states that upper bounds on the chromatic number are preserved under gluing along a fixed number of vertices.…”

Section: Introductionmentioning

confidence: 99%

“…The proof from [2] relies on an ad hoc induction hypothesis, which roughly states that upper bounds on the chromatic number are preserved under gluing along a fixed number of vertices.…”

Section: Introductionmentioning

confidence: 99%

Isolating Highly Connected Induced Subgraphs

Penev¹,

Thomassé²,

Trotignon³

2016

SIAM J. Discrete Math.

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We prove that any graph G of minimum degree greater than 2k2 −1 has a (k + 1)-connected induced subgraph H such that the number of vertices of H that have neighbors outside of H is at most 2k 2 − 1. This generalizes a classical result of Mader, which states that a high minimum degree implies the existence of a highly connected subgraph. We give several variants of our result, and for each of these variants, we give asymptotics for the bounds. We also we compute optimal values for the case when k = 2.Alon, Kleitman, Saks, Seymour, and Thomassen proved that in a graph of high chromatic number, there exists an induced subgraph of high connectivity and high chromatic number. We give a new proof of this theorem with a better bound.

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Amalgams and χ-Boundedness

Penev

2016

J. Graph Theory

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A class of graphs is hereditary if it is closed under isomorphism and induced subgraphs. A class scriptG of graphs is χ‐bounded if there exists a function f:N→N such that for all graphs G∈G, and all induced subgraphs H of G, we have that χ(H)≤f(ω(H)). We prove that proper homogeneous sets, clique‐cutsets, and amalgams together preserve χ‐boundedness. More precisely, we show that if scriptG and G* are hereditary classes of graphs such that scriptG is χ‐bounded, and such that every graph in G* either belongs to scriptG or admits a proper homogeneous set, a clique‐cutset, or an amalgam, then the class G* is χ‐bounded. This generalizes a result of [J Combin Theory Ser B 103(5) (2013), 567–586], which states that proper homogeneous sets and clique‐cutsets together preserve χ‐boundedness, as well as a result of [European J Combin 33(4) (2012), 679–683], which states that 1‐joins preserve χ‐boundedness. The house is the complement of the four‐edge path. As an application of our result and of the decomposition theorem for “cap‐free” graphs from [J Graph Theory 30(4) (1999), 289–308], we obtain that if G is a graph that does not contain any subdivision of the house as an induced subgraph, then χ(G)≤3ω(G)−1.

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Substitution and χ -boundedness

Cited by 30 publications

References 22 publications

χ‐bounds, operations, and chords

χ‐bounds, operations, and chords

Isolating Highly Connected Induced Subgraphs

Amalgams and χ-Boundedness

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