Results 1  10
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16
Complexes of graph homomorphisms
 Israel J. Math
"... Abstract. Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom ..."
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Cited by 33 (11 self)
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Abstract. Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom (Km, Kn) is homotopy equivalent to a wedge of (n−m)dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph G, and integers m ≥ 2 and k ≥ −1, we have ̟k 1 (Hom (Km, G)) ̸ = 0, then χ(G) ≥ k + m; here Z2action is induced by the swapping of two vertices in Km, and ̟1 is the first StiefelWhitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of Hom (G, H) induces a homotopy equivalence. It then follows that Hom (F, Kn) is homotopy equivalent to a direct product of (n−2)dimensional spheres, whileHom (F, Kn) is homotopy equivalent to a wedge of spheres, where F is an arbitrary forest and F is its complement. 1.1. Definition of the main object. 1.
Proof of the Lovász conjecture
 Ann. of Math
"... Abstract. In this paper we prove the Lovász Conjecture: If Hom(C2r+1, H) is kconnected, then χ(H) ≥ k + 4, where H is a finite undirected graph, C2r+1 is a cycle with 2r+1 vertices, r, k ∈ Z, r ≥ 1, k ≥ −1, and Hom(G, H) is the cell complex with the vertex set being the set of all graph homomorphi ..."
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Cited by 14 (1 self)
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Abstract. In this paper we prove the Lovász Conjecture: If Hom(C2r+1, H) is kconnected, then χ(H) ≥ k + 4, where H is a finite undirected graph, C2r+1 is a cycle with 2r+1 vertices, r, k ∈ Z, r ≥ 1, k ≥ −1, and Hom(G, H) is the cell complex with the vertex set being the set of all graph homomorphisms from G to H, and cells all allowed list Hcolorings of G. Our method is to compute, by means of spectral sequences, the obstructions to graph colorings, which lie either directly in the cohomology groups of Hom(C2r+1, Kn), or in the vanishing of the certain powers of StiefelWhitney classes of Hom(C2r+1, Kn), viewed as Z2spaces, resulting in proving even sharper statements.
Local chromatic number, Ky Fan’s theorem, and circular colorings
 Combinatorica
"... The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex colorcritical sub ..."
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Cited by 12 (5 self)
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The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex colorcritical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [14] which generalizes the BorsukUlam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2 ⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [15].) This yields a lower bound of ⌈t/2 ⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier
Topological obstructions to graph colorings
 Electron. Res. Announc. Amer. Math. Soc
"... Abstract. For any two graphs G and H Lovász has defined a cell complex Hom(G, H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with G ..."
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Cited by 7 (1 self)
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Abstract. For any two graphs G and H Lovász has defined a cell complex Hom(G, H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with G a cycle of odd length. More specifically, we show that If Hom(C2r+1, G) is kconnected, then χ(G)≥k + 4. Our actual statement is somewhat sharper, as we find obstructions already in the nonvanishing of powers of certain StiefelWhitney classes. 1.
THE BORSUKULAMPROPERTY, TUCKERPROPERTY AND CONSTRUCTIVE PROOFS IN COMBINATORICS
, 2005
"... Abstract. This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tuckerproperty of a finite group G is introduced and its relation to the topological BorsukUlampr ..."
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Cited by 2 (0 self)
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Abstract. This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tuckerproperty of a finite group G is introduced and its relation to the topological BorsukUlamproperty is discussed. Applications of the Tuckerproperty in combinatorics are demonstrated. 1.
On generalized Kneser hypergraph colorings
, 2006
"... In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs KGr sS, “generalized runiform Kneser hypergraphs with intersection multiplicities s. ” It generalized previous lower bounds by Kˇríˇz (1992/2000) for the case s = (1,...,1) without intersection mul ..."
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Cited by 2 (1 self)
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In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs KGr sS, “generalized runiform Kneser hypergraphs with intersection multiplicities s. ” It generalized previous lower bounds by Kˇríˇz (1992/2000) for the case s = (1,...,1) without intersection multiplicities, and by Sarkaria (1990) for S = ([n]) k. Here we discuss subtleties and difficulties that arise for intersection multiplicities si> 1: 1. In the presence of intersection multiplicities, there are two different versions of a “Kneser hypergraph,”depending on whether one admits hypergraph edges that are multisets rather than sets. We show that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset version. 2. The reductions to the case of prime r in the proofs by Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of r. Currently we have no valid proof for the lower bound result in the other cases. We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs.
HYPERGRAPHS WITH MANY KNESER COLORINGS
"... Abstract. For fixed positive integers r, k and ℓ with 1 ≤ ℓ < r and an runiform hypergraph H, let κ(H, k, ℓ) denote the number of kcolorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least ℓ elements. Consider the function KC(n, r, k, ℓ) = ..."
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Cited by 1 (1 self)
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Abstract. For fixed positive integers r, k and ℓ with 1 ≤ ℓ < r and an runiform hypergraph H, let κ(H, k, ℓ) denote the number of kcolorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least ℓ elements. Consider the function KC(n, r, k, ℓ) = maxH∈Hn κ(H, k, ℓ), where the maximum runs over the family Hn of all runiform hypergraphs on n vertices. In this paper, we determine the asymptotic behavior of the function KC(n, r, k, ℓ) for every fixed r, k and ℓ and describe the extremal hypergraphs. This variant of a problem of Erdős and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdős–Ko–Rado Theorem [7] on intersecting systems of sets. 1.
Colinear coloring on graphs
 In 3rd Annual Workshop on Algorithms and Computation (WALCOM’09
, 2009
"... Abstract. Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any g ..."
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Cited by 1 (1 self)
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Abstract. Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, andshowthatG can be colinearly colored in polynomial time by proposing a simple algorithm. The colinear coloring of a graph G is a vertex coloring such that two vertices can be assigned the same color, if their corresponding clique sets are associated by the set inclusion relation (a clique set of a vertex u is the set of all maximal cliques containing u); the colinear chromatic number λ(G) of G is the least integer k for which G admits a colinear coloring with k colors. Based on the colinear coloring, we define the χcolinear and αcolinear properties and characterize known graph classes in terms of these properties.
STABLE KNESER HYPERGRAPHS AND IDEALS IN N WITH THE NIKOD ´YM PROPERTY
"... Abstract. We use stable Kneser hypergraphs to construct ideals in N which are not nonatomic yet have the Nikod´ym property. 1. ..."
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Abstract. We use stable Kneser hypergraphs to construct ideals in N which are not nonatomic yet have the Nikod´ym property. 1.