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111
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.
Quadratic forms on graphs
 Invent. Math
, 2005
"... We introduce a new graph parameter, called the Grothendieck constant of a graph G = (V, E), which is defined as the least constant K such that for every A: E → R, ..."
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Cited by 32 (10 self)
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We introduce a new graph parameter, called the Grothendieck constant of a graph G = (V, E), which is defined as the least constant K such that for every A: E → R,
The chromatic number of Kneser hypergraphs
 Trans. Amer. Math. Soc
, 1986
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 28 (3 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Repeated communication and Ramsey graphs
 IEEE Transactions on Information Theory
, 1995
"... We study the savings afforded by repeated use in two zeroerror communication problems. We show that for some random sources, communicating one instance requires arbitrarilymany bits, but communicating multiple instances requires roughly one bit per instance. We also exhibit sources where the numbe ..."
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Cited by 27 (14 self)
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We study the savings afforded by repeated use in two zeroerror communication problems. We show that for some random sources, communicating one instance requires arbitrarilymany bits, but communicating multiple instances requires roughly one bit per instance. We also exhibit sources where the number of bits required for a single instance is comparable to the source’s size, but two instances require only a logarithmic number of additional bits. We relate this problem to that of communicating information over a channel. Known results imply that some channels can communicate exponentially more bits in two uses than they can in one use. 1
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 22 (11 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
Generalized Kneser coloring theorems with combinatorial proofs
 INVENTIONES MATH
, 2001
"... The Kneser conjecture (1955) was proved by Lovasz (1978) using the BorsukUlam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the BorsukUlam theorem and its extensions. Only in 2000, Matousek provided the rst combinatorial proof of t ..."
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Cited by 21 (4 self)
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The Kneser conjecture (1955) was proved by Lovasz (1978) using the BorsukUlam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the BorsukUlam theorem and its extensions. Only in 2000, Matousek provided the rst combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, AlonFranklLovasz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.
The Hardness of 3Uniform Hypergraph Coloring
 In Proc. of the 43rd Annual IEEE Symposium on Foundations of Computer Science
, 2002
"... We prove that coloring a 3uniform 2colorable hypergraph with any constant number of colors is NPhard. The best known algorithm [20] colors such a graph using O(n ) colors. Our result immediately implies that for any constants k > 2 and c 2 > c 1 > 1, coloring a kuniform c 1 colorable hype ..."
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Cited by 20 (4 self)
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We prove that coloring a 3uniform 2colorable hypergraph with any constant number of colors is NPhard. The best known algorithm [20] colors such a graph using O(n ) colors. Our result immediately implies that for any constants k > 2 and c 2 > c 1 > 1, coloring a kuniform c 1 colorable hypergraph with c 2 colors is NPhard; leaving completely open only the k = 2 graph case.
Transversal Numbers for Hypergraphs Arising in Geometry
 Adv. Appl. Math
, 2001
"... Introduction Helly's theorem asserts that if F is a finite family of convex sets in R d in which every d + 1 or fewer sets have a point in common then T F 6= ;. Our starting point, the (p; q) theorem, is a deep extension of Helly's theorem. It was conjectured by Hadwiger and Debrunner and prove ..."
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Cited by 16 (3 self)
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Introduction Helly's theorem asserts that if F is a finite family of convex sets in R d in which every d + 1 or fewer sets have a point in common then T F 6= ;. Our starting point, the (p; q) theorem, is a deep extension of Helly's theorem. It was conjectured by Hadwiger and Debrunner and proved by Alon and Kleitman [3]. Let p q 2 be integers. A family F of convex sets in R d is said to have the (p; q) property if among every p sets of F , some q have a point in common. Theorem 1 ((p; q) theorem, Alon & Kleitmen) For every p q d+ 1 there exists a number C =