On generalized Kneser hypergraph colorings

In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs \KG{r}{\pmb s}{\calS}, "generalized $r$-uniform Kneser hypergraphs with intersection multiplicities $\pmb s$." It generalized previous lower bounds by Kriz (1992/2000) for the case ${\pmb s}=(1,...,1)$ without intersection multiplicities, and by Sarkaria (1990) for \calS=\tbinom{[n]}k. Here we discuss subtleties and difficulties that arise for intersection multiplicities $s_i>1$: 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 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.Comment: 9 pages; added examples in Section 2; added reference ([11]), corrected minor typos; to appear in J. Combinatorial Theory, Series

A complexity dichotomy for hypergraph partition functions

We consider the complexity of counting homomorphisms from an $r$-uniform hypergraph $G$ to a symmetric $r$-ary relation $H$. We give a dichotomy theorem for $r>2$, showing for which $H$ this problem is in FP and for which $H$ it is #P-complete. This generalises a theorem of Dyer and Greenhill (2000) for the case $r=2$, which corresponds to counting graph homomorphisms. Our dichotomy theorem extends to the case in which the relation $H$ is weighted, and the goal is to compute the \emph{partition function}, which is the sum of weights of the homomorphisms. This problem is motivated by statistical physics, where it arises as computing the partition function for particle models in which certain combinations of $r$ sites interact symmetrically. In the weighted case, our dichotomy theorem generalises a result of Bulatov and Grohe (2005) for graphs, where $r=2$. When $r=2$, the polynomial time cases of the dichotomy correspond simply to rank-1 weights. Surprisingly, for all $r>2$ the polynomial time cases of the dichotomy have rather more structure. It turns out that the weights must be superimposed on a combinatorial structure defined by solutions of an equation over an Abelian group. Our result also gives a dichotomy for a closely related constraint satisfaction problem.Comment: 21 page

On the Chromatic Thresholds of Hypergraphs

Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. {\L}uczak and Thomass\'e recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Tur\'an number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Tur\'an number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemer\'edi for graphs and might be of independent interest. Many open problems remain.Comment: 37 pages, 4 figure

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs $G$ and $H$, we consider the polyhedral complex $\text{Hom}(G,H)$ that parametrizes the directed graph homomorphisms $f: G \rightarrow H$. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals. We study examples of directed Hom complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of $\text{Hom}(G,T_n)$ for $T_n$ a tournament. Finally, we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified statements and proofs, other corrections and minor revisions incorporating comments from referee