The chromatic number of almost stable Kneser hypergraphs

Meunier, Frédéric

doi:10.1016/j.jcta.2011.02.010

Cited by 44 publications

(82 citation statements)

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“…There are also various generalizations of Tucker's lemma. The next lemma is a combinatorial variant of the Z p -Tucker lemma proved and modified in [35] and [24], respectively.…”

Section: Tucker's Lemma and Its Generalizationsmentioning

confidence: 99%

“…As an approach to Conjecture A, Meunier [24] showed that χ(KG r (n, k) ∼ 2−stab ) = n−r(k−1) r−1 and he strengthened the above conjecture as follows.…”

Section: Introductionmentioning

confidence: 95%

“…The problem of finding a lower bound for the chromatic number of hypergraphs has been studied in the literature, see [7,18,19,28,29,35,36]. As in [24,35], our proofs rely on the Z p -Tucker lemma, which is a combinatorial generalization of the Borsuk-Ulam theorem. This lemma has also been used to simplify the evaluation of the chromatic number for some families of hypergraphs, e.g., almost 2-stable Kneser hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

“…This lemma has also been used to simplify the evaluation of the chromatic number for some families of hypergraphs, e.g., almost 2-stable Kneser hypergraphs. Like the Borsuk-Ulam theorem, the Z p -Tucker lemma is used to obtain lower bounds for the chromatic number of hypergraphs, see [24,35].…”

Section: Introductionmentioning

confidence: 99%

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On the chromatic number of general Kneser hypergraphs

Alishahi

Hajiabolhassan²

2015

Journal of Combinatorial Theory, Series B

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In a break-through paper, Lovász [20] determined the chromatic number of Kneser graphs. This was improved by Schrijver [27], by introducing the Schrijver subgraphs of Kneser graphs and showing that their chromatic number is the same as that of Kneser graphs. Alon, Frankl, and Lovász [2] extended Lovász's result to the usual Kneser hypergraphs and one of our main results is to extend this to a new family of general Kneser hypergraphs. Moreover, as a special case, we settle a question from Naserasr and Tardif [26]. In 2011, Meunier introduced almost 2-stable Kneser hypergraphs and determined their chromatic number as an approach to a supposition of Ziegler [35] and a conjecture of Alon, Drewnowski, and Łuczak [3]. In this work, our second main result is to improve this by computing the chromatic number of a large family of Schrijver hypergraphs. Our last main result is to prove the existence of a completely multicolored complete bipartite graph in every coloring of a graph which extends a result of Simonyi and Tardos [29]. JID:YJCTB AID:2930 /FLA [m1L; v1.156; Prn:10/06/2015; 9:53] P.2 (1-24) 2 M. Alishahi, H. Hajiabolhassan / J. Combin. Theory Ser. B ••• (••••) •••-••• The first two results are proved using a new improvement of the Dol'nikov-Kříž [7,18] bound on the chromatic number of general Kneser hypergraphs.

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“…There are also various generalizations of Tucker's lemma. The next lemma is a combinatorial variant of the Z p -Tucker lemma proved and modified in [35] and [24], respectively.…”

Section: Tucker's Lemma and Its Generalizationsmentioning

confidence: 99%

“…As an approach to Conjecture A, Meunier [24] showed that χ(KG r (n, k) ∼ 2−stab ) = n−r(k−1) r−1 and he strengthened the above conjecture as follows.…”

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the chromatic number of general Kneser hypergraphs

Alishahi

Hajiabolhassan²

2015

Journal of Combinatorial Theory, Series B

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“…Indeed, Dold's theorem implies that if there is such a map λ, then m ≥ n. It is worth noting that the idea of using Dold's theorem or some of it specializations such as the Z p -Tucker lemma has been used in several articles initiated by a fascinating paper of Matoušek [17]. For instance, see [1,4,6,7,12,18,19,24]. Usually, the most challenging task in using Dold's theorem is how to define the map λ, especially the sign part s(X).…”

Section: Notations and Toolsmentioning

confidence: 99%

Coloring properties of categorical product of general Kneser hypergraphs

2018

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More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture was generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier, in 2016, introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful type results for the coloring of categorical product of general Kneser hypergraphs, which generalize the Hajiabolhassan-Meunier result. Also, we present a new lower bound for the chromatic number of categorical product of general Kneser hypergraphs which can be much better than the Hajiabolhassan-Meunier lower bound. Using this lower bound, we enrich the family of hypergraphs satisfying Zhu's conjecture.

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On the Multichromatic Number of s‐Stable Kneser Graphs

Chen

2014

Journal of Graph Theory

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Abstract:For positive integers n and s, a subset S ⊆ [n] is s-stable if s ≤ |i − j | ≤ n − s for distinct i, j ∈ S . The s-stable r-uniform Kneser hypergraph K G r (n, k ) s-st abl e is the r-uniform hypergraph that has the collection of all s-stable k-element subsets of [n] as vertex set and whose edges are formed by the r-tuples of disjoint s-stable k-element subsets of [n]. Meunier (2011) conjectured that for positive integers n, k, r, s with k ≥ 2, s ≥ r ≥ 2, and n ≥ sk, the chromatic number of s-stable r -uniform Kneser hypergraphs is equal to (KG(n, k)) for n ≥ 2k . They conjectured that χ k (μ(KG(n, k))) = χ k (KG n, k ) + k for n ≥ 3k − 1. The case k = 1 was proved by Mycielski (1955). Lin et al. (2010) confirmed their conjecture for k = 2, 3, or when n is a multiple of k or n ≥ 3k 2 / ln k. In this paper, we investigate the multichromatic number of the usual s -stable Kneser graphs K G 2 (n, k ) s-st abl e . With the help of Fan's (1952) combinatorial lemma, we show that Meunier's conjecture is true for r is a power Journal of Graph Theory C 2014 Wiley Periodicals, Inc. 233234 JOURNAL OF GRAPH THEORY of 2 and s is a multiple of r, and Lin-Liu-Zhu's conjecture is true for n ≥ 3k.

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The chromatic number of almost stable Kneser hypergraphs

Cited by 44 publications

References 14 publications

On the chromatic number of general Kneser hypergraphs

On the chromatic number of general Kneser hypergraphs

Coloring properties of categorical product of general Kneser hypergraphs

On the Multichromatic Number of s‐Stable Kneser Graphs

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