New Construction of Graphs with High Chromatic Number and Small Clique Number

Daneshpajouh, Hamid Reza

doi:10.1007/s00454-017-9934-3

Cited by 6 publications

(7 citation statements)

References 18 publications

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“…A nice application of G-Borsuk graphs, is that they can produce graphs with large chromatic number and prescribed clique number. A similar construction using G-actions can also be found in [Dan18].…”

Section: Definitions and Resultsmentioning

confidence: 96%

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Generalized Borsuk Graphs

Martinez-Figueroa

2021

Preprint

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Given a finite group G acting freely on a compact metric space M , and ε > 0, we define the G-Borsuk graph on M by drawing edges x ∼ y whenever there is a non-identity g ∈ G such that d (x, gy) ≤ ε. We show that when ε is small, its chromatic number is determined by the topology of M via its G-covering number, which is the minimum k such that there is a closed coverWe are interested in bounding this number. We give lower bounds using G-actions on Hom-complexes, and upper bounds using a recursive formula on the dimension of M . We conjecture that the true chromatic number coincides with the lower bound, and give computational evidence. We also study random G-Borsuk graphs, which are random induced subgraphs. For these, we compute thresholds for ε that guarantee that the chromatic number is still that of the whole G-Borsuk graph. Our results are tight (up to a constant) when the G-index and dimension of M coincide.

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Section: Definitions and Resultsmentioning

confidence: 96%

“…This result is topological, by bounding the G-index of the cellular structure of Hom(K m , H), following ideas of [Lov78], [BK03] and [BK06]. It is worth noting, that similar G-actions on Hom-Posets have been independently studied in [Dan18]. We start by restating the result we will prove.…”

Section: Lower Boundmentioning

confidence: 84%

Generalized Borsuk Graphs

Martinez-Figueroa

2021

Preprint

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“…To propose a new method for finding topological lower bounds for the chromatic numbers of graphs, compatibility graphs were introduced in [7]. Later, some other applications of this new family of graphs, such as a new proof of the well-known Kneser conjecture, and a new way of constructing graphs with high chromatic numbers and small clique numbers, have been found [6]. Moreover, in order to attack to a generalization of the famous Hedetniemi conjecture, a new version of this concept has been introduced for hypergraphs.…”

Section: Strong Compatibility Graphs and Dold's Theoremmentioning

confidence: 99%

“…We refer the interested reader to [10,13,28] for generalizations, and [1,9,10,20,22] for applications. Not long ago, a new generalization of octahedral Tucker's lemma, called G-Tucker's lemma, was introduced in [6]. Moreover, as an application of that generalization, a new method for constructing graphs with high chromatic number and small clique number was given.…”

Section: Introductionmentioning

confidence: 99%

Dold’s theorem from viewpoint of strong compatibility graphs

Daneshpajouh

2020

European Journal of Combinatorics

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Let G be a non-trivial finite group. The well-known Dold's theorem states that: There is no continuous G-equivariant map from an n-connected simplicial G-complex to a free simplicial G-complex of dimension at most n. In this paper, we give a new generalization of Dold's theorem, by replacing "dimension at most n" with a sharper combinatorial parameter. Indeed, this parameter is the chromatic number of a new family of graphs, called strong compatibility graphs, associated to the target space. Moreover, in a series of examples, we will see that one can hope to infer much more information from this generalization than ordinary Dold's theorem. In particular, we show that this new parameter is significantly better than the dimension of target space "for almost all free Z 2 -simplicial complex." In addition, some other applications of strong compatibility graphs will be presented as well.In particular, a new way for constructing triangle-free graphs with high chromatic numbers from an n-sphere S n , and some new results on the limitations of topological methods for determining the chromatic number of graphs will be given.

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“…After about two decades, this conjecture was confirmed by Lovász [12] via a tool from algebraic topology! Since then many other proofs have been found, see for instance [6,10,11,13]. Later, a generalization of Kneser's conjecture was raised by Erdős [8].…”

Section: Introductionmentioning

confidence: 97%

On the chromatic number of generalized Kneser hypergraphs

Daneshpajouh

2019

European Journal of Combinatorics

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The generalized Kneser hypergraph KG r (n, k, s) is the hypergraph whose vertices are all the k-subsets of {1, . . . , n}, and edges are r-tuples of distinct vertices such that any pair of them has at most s elements in their intersection. In this note, we show that for each non-negative integers k, n, r, s satisfying n ≥ r(k − 1) + 1, k > s ≥ 0, and r ≥ 2, we havewhich extends the previously known result by Alon-Frankl-Lovász.

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New Construction of Graphs with High Chromatic Number and Small Clique Number

Abstract: In this note, we introduce a new method for constructing graphs with high chromatic number and small clique. Indeed, via this method, we present a new proof for the well-known Kneser's conjecture.

Cited by 6 publications

References 18 publications

Generalized Borsuk Graphs

Generalized Borsuk Graphs

Dold’s theorem from viewpoint of strong compatibility graphs

On the chromatic number of generalized Kneser hypergraphs

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