Neighborhood Complexes of Stable Kneser Graphs

Björner, Anders; Longueville, Mark de

doi:10.1007/s00493-003-0012-5

Cited by 23 publications

(34 citation statements)

References 5 publications

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“…Note that this kind of edges has already been considered and named quasistable in a paper by Björner and de Longueville [3].…”

Section: Resultsmentioning

confidence: 92%

The chromatic number of almost stable Kneser hypergraphs

Meunier

2011

Journal of Combinatorial Theory, Series A

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International audienceLet V(n,k,s) be the set of k-subsets S of [n] such that for all i,j∈S, we have |i−j|⩾s. We define almost s-stable Kneser hypergraph View the MathML source to be the r-uniform hypergraph whose vertex set is V(n,k,s) and whose edges are the r-tuples of disjoint elements of V(n,k,s). With the help of a Zp-Tucker lemma, we prove that, for p prime and for any n⩾kp, the chromatic number of almost 2-stable Kneser hypergraphs View the MathML source is equal to the chromatic number of the usual Kneser hypergraphs View the MathML source, namely that it is equal to View the MathML source. Related results are also proved, in particular, a short combinatorial proof of Schrijverʼs theorem (about the chromatic number of stable Kneser graphs) and some evidences are given for a new conjecture concerning the chromatic number of usual s-stable r-uniform Kneser hypergraphs

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“…Note that this kind of edges has already been considered and named quasistable in a paper by Björner and de Longueville [3].…”

Section: Resultsmentioning

confidence: 92%

The chromatic number of almost stable Kneser hypergraphs

Meunier

2011

Journal of Combinatorial Theory, Series A

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“…The box complexes of Schrijver graphs are tidy as well (spheres up to homotopy [3]). This means that one can prove Kneser's conjecture using Sarkaria's bound (or any higher suspension of the box complex).…”

Section: Remark 32mentioning

confidence: 99%

Homotopy types of box complexes

Csorba

2007

Combinatorica

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In [14] Matoušek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovász's original bound [9] can be restated as χ (G) ≥ ind(B(G)) + 2. Sarkaria's bound [15] can be formulated as χ (G) ≥ ind(B0(G)) + 1. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. The most interesting result is that up to Z2-homotopy the box complex B(G) can be any Z2-space. This together with topological constructions allows us to construct graphs showing that the mentioned two bounds are different. Some of the results were announced in [14].

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“…We have not seen this connection made explicit in the literature, even though Ziegler [24] combined ideas from both proofs and work by Matoušek [17] in a combinatorial proof of χ (SG n,k ) = k + 2 and even though the vertex criticality of stable Kneser graphs had suggested that the neighbourhood complex N (SG n,k ) is homotopy equivalent to a k-sphere, which was proved by Björner and de Longueville [5].…”

Section: Detailed Overview and Resultsmentioning

confidence: 99%

“…The construction of Section 3 leads to a more conceptual proof of the homotopy equivalence N (SG n,k ) S k of [5] and, using the result of Section 4, also of the following D 2m -equivariant version.…”

Section: Homotopymentioning

confidence: 99%

The equivariant topology of stable Kneser graphs

Schultz

2011

Journal of Combinatorial Theory, Series A

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The stable Kneser graph SG n,k , n 1, k 0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k + 2, its vertices are certain subsets of a set of cardinality m = 2n + k. (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K 2 , SG n,k ) S k . Björner and de LonguevilleThe dihedral group D 2m acts canonically on SG n,k , the group C 2 with 2 elements acts on K 2 . We almost determine the (C 2 × D 2m )-homotopy type of Hom(K 2 , SG n,k ) and use this to prove the following results. The graphs SG 2s,4 are homotopy test graphs, i.e. for every graph H and r 0 such that Hom(SG 2s,4 , H) is (r − 1)-connected, the chromatic number χ (H) is at least r + 6.If k / ∈ {0, 1, 2, 4, 8} and n N(k) then SG n,k is not a homotopy test graph, i.e. there are a graph G and an r 1 such that Hom(SG n,k , G) is (r − 1)-connected and χ (G) < r + k + 2.

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Neighborhood Complexes of Stable Kneser Graphs

Cited by 23 publications

References 5 publications

The chromatic number of almost stable Kneser hypergraphs

The chromatic number of almost stable Kneser hypergraphs

Homotopy types of box complexes

The equivariant topology of stable Kneser graphs

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