We prove that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard. 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 k-uniform c 1 -colorable hypergraph with c 2 colors is NP-hard; leaving completely open only the k = 2 graph case.

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 #P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1

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 color-critical 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 Borsuk-Ulam 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

We estimate the chromatic number of graphs whose vertex set is the set of edges of a complete geometric graph on n points, and adjacency is defined in terms of geometric disjointness or geometric intersection.

We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovász. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of highly connected, highly symmetric manifolds. 1

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.

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 Tucker-property of a finite group G is introduced and its relation to the topological Borsuk-Ulam-property is discussed. Applications of the Tucker-property in combinatorics are demonstrated. 1.

L. Lovász has shown in [9] that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature (Theorem 3.1). Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C m,r. 1

We investigate some coloring properties of Kneser graphs. A star-free coloring is a proper coloring c: V (G) → N such that no path with three vertices may be colored with just two consecutive numbers. The minimum positive integer t for which there exists a star-free coloring c: V (G) → {1, 2,..., t} is called the starfree chromatic number of G and denoted by χs(G). In view of Tucker-Ky Fan’s lemma, we show that for any Kneser graph KG(n, k) we have χs(KG(n, k)) ≥ max{2χ(KG(n, k)) − 10, χ(KG(n, k))} where n ≥ 2k ≥ 4. Moreover, we show that χs(KG(n, k)) = 2χ(KG(n, k)) − 2 = 2n − 4k + 2 provided that n ≤ 8 3 k. This gives a partial answer to a conjecture of [12]. Also, we conjecture that for any positive integers n ≥ 2k ≥ 4 we have χs(KG(n, k)) = 2χ(KG(n, k)) − 2.