In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph $\mbox{KG}^r_s(n,k)$, has all $k$-subsets of $\{1,\dots,n\}$ as the vertex set and all multi-sets $\{A_1,\dots, A_r\}$ of $k$-subsets with $s$-wise empty intersections as the edge set. The case $r=s=2$, was considers by Kneser \cite{K} in 1955, where he conjectured that its chromatic number is $n-2(k-1)$. This was finally proved by Lovász \cite{L} in 1978. The case $r>2$ and $s=2$, was considered by Erdös in 1973, and he conjectured that its chromatic number is $\left\lceil\frac{n-r(k-1)}{r-1}\right\rceil$. This conjecture was proved by Alon, Frankl and Lovász \cite{AFL} in 1986. The case where $s>2$, was considered by Sarkaria \cite{S} in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler \cite{Z'} in 2006 in the induction method of Sarkaria on the number of prime factors of $r$, and Sarkaria's proof only worked when $s$ is less than the smallest prime factor of $r$ or $s=2$. In this paper, by applying the $\mathbb Z_p$-Tucker lemma of Ziegler \cite{Z} and Meunier \cite{M}, we finally prove the general Erdös conjecture and prove the claimed result of Sarkaria for any $2\le s\le r$. We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lovász \cite{AFL} and compute the connectivity of certain simplicial complexes that might be of interest in their own right.
A general Kneser hypergraph ${\rm KG}^r(\mathcal{H})$ is an $r$-uniform hypergraph that somehow encodes the edge intersections of a ground hypergraph $\mathcal{H}$. The colorability defect of $\mathcal{H}$ is a combinatorial parameter providing a lower bound for the chromatic number of ${\rm KG}^r(\mathcal{H})$ which is addressed in a series of works by Dol'nikov [Sibirskii Matematicheskii Zhurnal, 1988}], Kř\'ıž [Transaction of the American Mathematical Society, 1992], and Ziegler~[Inventiones Mathematicae, 2002]. In this paper, we define a new combinatorial parameter, the equitable colorabil…
In this paper among some other results and by using the existance of Steiner triple systems, we determine the total dominator chromatic number of the Kneser graph KG(n,2).
In this note, we prove that for any integer $n\geq 3$ the b-chromatic number of the Kneser graph $KG(m,n)$ is greater than or equal to $2{\lfloor {m\over 2} \rfloor \choose n}$. This gives an affirmative answer to a conjecture of [6].
We determine the chromatic number of some graphs of flags in buildings of type $A_4$, namely of the Kneser graphs of flags of type $\{2,4\}$ in the vector spaces $GF(q)^5$ for $q\geq3$, and of the Kneser graph of flags of type $\{2,3\}$ in the vector spaces $GF(q)^5$ for large $q$.
Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Habolhassan and Zhu [Circular chromatic number of Kneser graphs, Journal of Combinatorial Theory Series B, 2003] proved that if $n$ is sufficiently large with respect to $k$, then the Schrijver graph ${\rm SG}(n,k)$ has the same chromatic and circular chromatic number. Later, Meunier [A topological lower bound for the circular chromatic number of Schrijver graphs, Journal of Graph Theory, 2005] and independently, Simonyi and Tard…
The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The \emph{$(p,q)$-extremal problem} consists in finding the maximum number of edges on a $k$-uniform hypergraph $\mathcal{H}$ with $n$ vertices such that among any $p$ edges some $q$ of them have no empty intersection. In this paper we have found a link between the fractional chromatic number of Kneser hypergraphs and the $(p,q)$-extremal problem and also solve the $(…
Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijver graphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromatic number of random Kneser graphs $\KG_{n,k}(ρ)$ and proved that, in many cases, the chromatic numbers of the random Kneser graph $\KG_{n,k}(ρ)$ and the Kneser graph $\KG_{n,k}$ are almost surely closed. He also marked the studying of the chromatic number of random Kneser hypergraphs $\KG^r_{n,k}(ρ)$ as a very interesting problem. With the help of $\Z_p$-Tucker lemma, a combinatorial generalization of the Borsuk-Ulam theorem, we generalize Kupavskii'…
Kneser's 1955 conjecture -- proven by Lovász in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser's conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász' result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser's conjecture. Here we prove a common generalization of a…
In 2011, Meunier conjectured that for positive integers $n,k,r,s$ with $ k\geq 2$, $r\geq 2$, and $n\geq \max (\{r,s\})k$, the chromatic number of $s$ -stable $r$-uniform Kneser hypergraphs is equal to $\left\lceil \frac{n-\max (\{r,s\})(k-1)}{r-1}\right\rceil $. It is a strengthened version of the conjecture proposed by Ziegler (2002), and Alon, Drewnowski and Łuczak (2009). The problem about the chromatic number of almost $s$-stable $r$ -uniform Kneser hypergraphs has also been introduced by Meunier (2011). For the $r=2$ case of the Meunier conjecture, Jonsson (2012) provided a purely combin…
We obtain a new weak Hilton-Milner type result for intersecting families of $k$-spaces in $\mathbb{F}_q^{2k}$, which improves several known results. In particular the chromatic number of the $q$-Kneser graph $qK_{n:k}$ was previously known for $n > 2k$ (except for $n=2k+1$ and $q=2$) or $k < q \log q - q$. Our result determines the chromatic number of $qK_{2k:k}$ for $q \geq 5$, so that the only remaining open cases are $(n, k) = (2k, k)$ with $q \in \{ 2, 3, 4 \}$ and $(n, k) = (2k+1, k)$ with $q = 2$.
We determine the chromatic number of the Kneser graph qΓ_{7,{3,4}} of flags of vectorial type {3, 4} of a rank 7 vector space over the finite field GF(q) for large q and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all large independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs qΓ_{2d+1,{d,d+1}} of flags of vectorial type {d, d+1} of a rank 2d+1 vector space over GF(q) for large q as long as the large independent sets of…
Let $V(n,k,s)$ be the set of $k$-subsets $S$ of $[n]$ such that for all $i,j\in S$, we have $|i-j|\geq s$ We define almost $s$-stable Kneser hypergraph $KG^r{{[n]}\choose k}_{s{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ to be the $r$-uniform hypergraph whose vertex set is $V(n,k,s)$ and whose edges are the $r$-uples of disjoint elements of $V(n,k,s)$. With the help of a $Z_p$-Tucker lemma, we prove that, for $p$ prime and for any $n\geq kp$, the chromatic number of almost 2-stable Kneser hypergraphs $KG^p {{[n]}\choose k}_{2{\tiny{\textup{-stab}}}}^{\displaystyle\sim}$ is equal to the chroma…
The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser's conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant m…
The main purpose of this note is to give a counterexample to the following conjecture, raised by Florian Frick [\textit{Int. Math. Res. Not. IMRN 2020 (13), 4037-4061 (2020)}]. Conjecture. Let $r\geq 3$ and let $\mathcal{F}$ be a set system. Then $$χ\left(\textrm{KG}^r\left(\mathcal{F}_{r-stab}\right)\right)\geq\left\lceil\frac{cd_r\left(\mathcal{F}\right)}{r-1}\right\rceil.$$
Using a $Z_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a hypergraph).
In this paper, in view of $Z_p$-Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol'nikov-K{ř}{\'ı}{ž} bound. Next, we introduce multiple Kneser hypergraphs and we specify the chromatic number of some multiple Kneser hypergraphs. For a vector of positive integers $\vec{s}=(s_1,s_2,\ldots,s_m)$ and a partition $π=(P_1,P_2,\ldots,P_m)$ of $\{1,2,\ldots,n\}$, the multiple Kneser hypergraph ${\rm KG}^r(π; \vec{s};k)$ is a hypergraph with the vertex set $$V=\left\{A:\ A\subseteq P_1\cup P_2\cup\cdots \cup P_m,\ |A|=k, \forall 1\leq i\leq m;\ |A\cap…
The generalized Kneser hypergraph $KG^{r}(n,k,s)$ is the hypergraph whose vertices are all the $k$-subsets of $\{1,\ldots ,n\}$, and edges are $r$-tuples of distinct vertices such that any pair of them has at most $s$ elements in their intersection. In this note, we show that for each non-negative integers $k, n, r, s$ satisfying $n \geq r(k-1)+1$, $k > s\geq 0$, and $r\geq 2$, we have $$χ({KG}^{r}(n,k,s))\geq\left\lceil\frac{n-r(k-s-1)}{r-1}\right\rceil,$$ which improves the previously known result by Alon--Frankl--Lovász.
In 2002, A. Björner and M. de Longueville showed the neighborhood complex of the $2$-stable Kneser graph ${KG(n, k)}_{2-\textit{stab}}$ has the same homotopy type as the $(n-2k)$-sphere. A short time ago, an analogous result about the homotopy type of the neighborhood complex of almost $s$-stable Kneser graph has been announced by J. Osztényi. Combining this result with the famous Lovász's topological lower bound on the chromatic number of graphs has been yielded a new way for determining the chromatic number of these graphs which was determined a bit earlier by P. Chen. In this paper we prese…
More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture were generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier (2016) introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful t…
Let $c$ be a proper $k$-coloring of a connected graph $G$ and $Π=(C_1,C_2,...,C_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $Π$ is defined to be the ordered $k$-tuple $$c_{{}_Π}(v):=(d(v,C_1),d(v,C_2),...,d(v,C_k)),$$ where $d(v,C_i)=\min\{d(v,x) |x\in C_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $c$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $\Cchi_{{}_L}(G)$. In this …
Let $n\ge 1$ and $s\ge 1$ be integers. An almost $s$-stable subset $A$ of $[n]=\{1,\dots,n\}$ is a subset such that for any two distinct elements $i, j\in A$, one has $|i-j|\ge s$. For a family $\cal F$ of non-empty subsets of $[n]$ and an integer $r\ge 2$, the chromatic number of the $r$-uniform Kneser hypergraph $\mbox{KG}^r({\cal F})$, whose vertex set is $\cal F$ and whose edge set is the set of $\{A_1,\dots, A_r\}$ of pairwise disjoint elements in $\cal F$, has been studied extensively in the literature and Abyazi Sani and Alishahi were able to give a lower bound for it in terms of the eq…
Let $G(n, r, s)$ be a graph whose vertices are all $r$-element subsets of an $n$-element set, in which two vertices are adjacent if they intersect in exactly $s$ elements. In this paper we study chromatic numbers of $G(n, r, s)$ with $r, s$ being fixed constants and $n$ tending to infinity. Using a recent result of Keevash on existence of designs we deduce an inequality $χ(G(n, r, s)) \le (1+o(1))n^{r-s} \frac{(r-s-1)!}{(2r-2s-1)!}$ for $r > s$ with $r, s$ fixed constants. This inequality gives sharp upper bounds for $r \le 2s+1$. Also we develop an elementary approach to this problem and prov…
Combining Ky Fan's theorem with ideas of Greene and Matousek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
The local chromatic number of a graph was introduced by Erdos et al. in 1986. 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 which generalizes the Borsuk-Ulam theo…
Let $Γ$ be the graph whose vertices are the chambers of the finite projective space $PG(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\ge 43$ we determine the largest independent set of $Γ$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\ge 47$, this information is then used to show that $Γ$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove simi…
Université de Paris（巴黎大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF研究所）；University of Waterloo（滑铁卢大学）；Department of Combinatorics and Optimization, University of Waterloo（滑铁卢大学组合优化系）；University of West Bohemia（西波希米亚大学）；NTIS (New Technologies for the Information Society)（新信息技术欧洲卓越中心）
This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lovász in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This conjecture stated that the \emph{Kneser graph} $\KG_{m,n}$, the graph with all $k$-element subsets of $\{1,2,...,n\}$ as vertices and all pairs of disjoint sets as edges, has chromatic number $n-2k+2$. Several other proofs have since been published (by Bárány, Schrijver, Dolnikov, Sarkaria, Kriz, Greene, and others), all of them based on some version of the Borsuk--U…
One of the most famous conjecture in graph theory is Hedetniemi's conjecture stating that the chromatic number of the categorical product of graphs is the minimum of their chromatic numbers. Using a suitable extension of the definition of the categorical product, Zhu proposed in 1992 a similar conjecture for hypergraphs. We prove that Zhu's conjecture is true for the usual Kneser hypergraphs of same rank. It provides to the best of our knowledge the first non-trivial and explicit family of hypergraphs with rank larger than two satisfying this conjecture (the rank two case being Hedetniemi's co…
In this short note, the purpose is to provide an upper bound for the b-chromatic number of Kneser graphs. Our bound improves the upper bound that was presented by Balakrishnan and Kavaskar in [b-coloring of Kneser graphs, Discrete Appl. Math. 160 (2012), 9-14].
There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi and Hajiabolhassan [{\it On the chromatic number of general Kneser hypergraphs, Journal of Combinatorial Theory, Series B, 2015}] introduced a lower bound for the chromatic number of Kneser hypergraphs ${\rm KG}^r({\mathcal H})$. Next, Meunier [{\it Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014}] improved their re…
Given positive integers $n\ge 2k$, the {\it Kneser graph} $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$-element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of the {\it random Kneser graph} $KG_{n,k}(p)$, that is, the graph obtained from $KG_{n,k}$ by including each of the edges of $KG_{n,k}$ independently and with probability $p$. …
A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lovász states that the chromatic number of a Kneser graph $KG_{n,k}$ is equal to $n-2k+2$. In this paper we study the chromatic number of a random subgraph of a Kneser graph $KG_{n,k}$ as $n$ grows. A random subgraph $KG_{n,k}(p)$ is obtained by including each edge of $KG_{n,k}$ with probability $p$. For a wide range of parameters $k = k(n), p = p(n)$ we show that $χ(KG_…
Let $n,k,r$ be positive integers with $n \geq rk$ and $r \geq 2$. Consider a circle $C$ with~$n$ points~$1,\ldots,n$ in clockwise order. The $r$-stable \emph{interlacing graph} $\text{IG}_{n,k}^{(r)}$ is the graph with vertices corresponding to $k$-subsets $S$ of $\{1,...,n\}$ such that any two distinct points in~$S$ have distance at least~$r$ around the circle, and edges between~$k$-subsets $P$ and $Q$ if they \emph{interlace}: after removing the points in~$P$ from $C$, the points in~$Q$ are in different connected components. In this paper we prove that the circular chromatic number of $\text…
The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-elements subsets of an $n$-element set, with two vertices adjacent if the sets are disjoint. The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Determining the chromatic number of the square of the Kneser graph $K(2k+1, k)$ is an interesting problem, but not much progress has been made. Kim and Nakprasit \cite{2004KN} showed that $χ(K^2(2k+1,k)) \leq 4k+2$, and Chen, Lih, and Wu \cite{2009CLW} showed that…
The Kneser graph $KG_{n,k}$ is the graph whose vertices are the $k$-element subsets of $[n],$ with two vertices adjacent if and only if the corresponding sets are disjoint. A famous result due to Lovász states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper we discuss the chromatic number of random Kneser graphs and hypergraphs. It was studied in two recent papers, one due to Kupavskii, who proposed the problem and studied the graph case, and the more recent one due to Alishahi and Hajiabolhassan. The authors of the latter paper had extended the result of Kupavskii …
The stable Kneser graph $SG_{n,k}$, $n\ge1$, $k\ge0$, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Björner and de Longueville \cite{anders-mark} have shown that its box complex is homotopy equivalent to a sphere, $\Hom(K_2,SG_{n,k})\homot\Sphere^k$. The 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 $(C_2\times D_{2m})$-homotopy type of $\Hom(K_2,SG_{n,k})$ and use this to prove the following resu…
We show that there are $k$ simple graphs whose Kronecker covers are isomorphic to the bipartite Kneser graph $H(n,k)$, and that their chromatic numbers coincide with $χ(K(n,k)) = n - 2k + 2$. We also determine the automorphism groups of these graphs.
Let $G$ be a graph and $r\in\mathbb{N}$. The matching Kneser graph $\textsf{KG}(G, rK_2)$ is a graph whose vertex set is the set of $r$-matchings in $G$ and two vertices are adjacent if their corresponding matchings are edge-disjoint. In [Alishahi, M. and Hajiabolhassan, H., On the Chromatic Number of Matching Kneser Graphs, Combin. Probab. and Comput. 29 (2020), no. 1, 1--21.] it was conjectured that for any connected graph $G$ and positive integer $r\geq 2$, the chromatic number of $\textsf{KG}(G, rK_2)$ is equal to $|E(G)|-\textsf{ex}(G,rK_2)$, where $\textsf{ex}(G,rK_2)$ denotes the larges…
Let $n\ge 1$, $r\ge 2$, and $s\ge 0$ be integers and ${\cal P}=\{P_1,\dots, P_l\}$ be a partition of $[n]=\{1,\dots, n\}$ with $|P_i|\le r$ for $i=1,\dots, l$. Also, let $\cal F$ be a family of non-empty subsets of $[n]$. The $r$-uniform Kneser-type hypergraph $\mbox{KG}^r({\cal F}, {\cal P},s)$ is the hypergraph with the vertex set of all $\cal P$-admissible elements $F\in {\cal F}$, that is $|F\cap P_i|\le 1$ for $i=1,\dots, l$ and the edge set of all $r$-subsets $\{F_1,\dots, F_r\}$ of the vertex set that $|F_i\cap F_j|\le s$ for all $1\le i<j\le r$. In this article, we extend the equitable…
In this note, we introduce a new method for constructing graphs with high chromatic number and small clique. Indeed, via this method, we present a new proof for the well-known Kneser's conjecture.
For integers n\geq 1, k\geq 0, the stable Kneser graph SG_{n,k} (also called the Schrijver graph) has as vertex set the stable n-subsets of [2n+k] and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not contain any 2-subset of the form {i,i+1} or {1,2n+k}. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number k+2. This article contains a study of the independence complexes of SG_{n,k} for small values of n and k. Our contributions a…
Hadwiger's conjecture asserts that every graph with chromatic number $t$ contains a complete minor of order $t$. Given integers $n \ge 2k+1 \ge 5$, the Kneser graph $K(n, k)$ is the graph with vertices the $k$-subsets of an $n$-set such that two vertices are adjacent if and only if the corresponding $k$-subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser graphs.
A Kneser representation KG(H) for a graph G is a bijective assignment of hyperedges of a hypergraph H to the vertices of G such that two vertices of G are adjacent if and only if the corresponding hyperedges are disjoint. In this paper, we introduce a colored version of the Turan number and use that to determine the chromatic number of some families of graphs in terms of the generalized Turan number of graphs. In particular, we determine the chromatic number of every Kneser multigraph KG(H), where the vertex set of H is the edge set of a multigraph G such that the multiplicity of each edge is …
For positive integers n and r we define the Haggkvist-Hell graph, H_{n:r}, to be the graph whose vertices are the ordered pairs (h,T) where T is an r-subset of [n], and h is an element of [n] not in T. Vertices (h_x,T_x) and (h_y,T_y) are adjacent iff h_x \in T_y, h_y \in T_x, and T_x and T_y are disjoint. These triangle-free arc transitive graphs are an extension of the idea of Kneser graphs, and there is a natural homomorphism from the Haggkvist-Hell graph, H_{n:r}, to the corresponding Kneser graph, K_{n:r}. Haggkvist and Hell introduced the r=3 case of these graphs, showing that a cubic gr…
The Kneser conjecture (1955) was proved by Lovász (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matoušek provided the first 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, Alon-Frankl-Lovász, Sarkaria, and Kriz. We also give a comb…
Lovasz's striking proof of Kneser's conjecture from 1978 using the Borsuk--Ulam theorem provides a lower bound on the chromatic number of a graph. We introduce the shore subdivision of simplicial complexes and use it to show an upper bound to this topological lower bound and to construct a strong Z_2-deformation retraction from the box complex (in the version introduced by Matousek and Ziegler) to the Lovasz complex. In the process, we analyze and clarify the combinatorics of the complexes involved and link their structure via several ``intermediate'' complexes.
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," d…
