Let $n$, $k$ and $t$ be integers with $1\leq t< k \leq n$. The \emph{generalized Kneser graph} $K(n,k,t)$ is a graph whose vertices are the $k$-subsets of a fixed $n$-set, where two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|<t$. The graph $K(n,k,1)$ is the well-known \emph{Kneser graph}. In 2014, Harvey and Wood determined the exact treewidth of the Kneser graphs for large $n$ with respect to $k$. In this paper, we give the exact treewidth of the generalized Kneser graphs for $t\geq2$ and large $n$ with respect to $k$ and $t$. In the special case when $t=k-1$, the graph $K(n,k,k-1)$ usually denoted by $\overline{J(n,k)}$ which is the complement of the Johnson graph $J(n,k)$. We give a more precise result for the exact value of the treewidth of $\overline{J(n,k)}$ for any $n$ and $k$.
Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n,k) is the graph with vertex set $\binom{[n]}{k}$, such that two vertices are adjacent if they are disjoint. We determine, for large values of n with respect to k, the exact treewidth of the Kneser graph. In the process of doing so, we also prove a strengthening of the Erdős-Ko-Rado Theorem (for large n with respect to k) when a number of disjoint pairs of k-sets are allowed.
Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$, where $q$ is a prime power. Define the \emph{generalized $q$-Kneser graph} $K_q(n,k,t)$ to be the graph whose vertices are the $k$-dimensional subspaces of $V$ and two vertices $F_1$ and $F_2$ are adjacent if $\dim(F_1\cap F_2)<t$. Then $K_q(n,k,1)$ is the well-known $q$-Kneser graph. In this paper, we determine the treewidth of $K_q(n,k,t)$ for $n\geq 2t(k-t+1)+k+1$ and $t\ge 1$ exactly. Note that $K_q(n,k,k-1)$ is the complement of the Grassmann graph $G_q(n,k)$. We give a more precise result for the treewidth of …
The generalized Kneser graph $K(n,k,t)$ for integers $k>t>0$ and $n>2k-t$ is the graph whose vertices are the $k$-subsets of $\{1,\dots,n\}$ with two vertices adjacent if and only if they share less than $t$ elements. We determine the treewidth of the generalized Kneser graphs $K(n,k,t)$ when $t\ge 2$ and $n$ is sufficiently large compared to $k$. The imposed bound on $n$ is a significant improvement of a previously known bound. One consequence of our result is the following. For each integer $c\ge 1$ there exists a constant $K(c)\ge 2c$ such that $k\ge K(c)$ implies for $t=k-c$ that $$tw(K(n,…
Rubei et. al., established results for the distance matrix of positive weighted Petersen graphs. Focusing on the properties of the distance matrix, we generalized positive weighted Petersen graphs results to Kneser graphs. We analyzed theorems established by Rubei et al. and used girth of the generalized Petersen graphs and Kneser graphs to conclude generalizations. Further, we examined the properties of positive weighted generalized Petersen graphs. We generalized the properties of distance matrices of positive weighted Petersen graphs to positive weighted generalized Petersen graphs.
We determine the chromatic number of some graphs of flags in buildings of type $A_4$, namely of the Kneser graphs of flags of type $\{2,4\}$ in the vector spaces $GF(q)^5$ for $q\geq3$, and of the Kneser graph of flags of type $\{2,3\}$ in the vector spaces $GF(q)^5$ for large $q$.
The generalized Kneser hypergraph $KG^{r}(n,k,s)$ is the hypergraph whose vertices are all the $k$-subsets of $\{1,\ldots ,n\}$, and edges are $r$-tuples of distinct vertices such that any pair of them has at most $s$ elements in their intersection. In this note, we show that for each non-negative integers $k, n, r, s$ satisfying $n \geq r(k-1)+1$, $k > s\geq 0$, and $r\geq 2$, we have $$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 chomp on graphs, two players alternatingly pick an edge or a vertex from a graph. The player that cannot move any more loses. The questions one wants to answer for a given graph are: Which player has a winning strategy? Can a explicit strategy be devised? We answer these questions (and determine the Nim-value) for the class of generalized Kneser graphs and for several families of Johnson graphs. We also generalize some of these results to the clique complexes of these graphs. Furthermore, we determine which player has a winning strategy for some classes of threshold graphs.
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…
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…
Alon, Frankl, and Lovász proved a conjecture of Erdős that one needs at least $\lceil \frac{n-r(k-1)}{r-1} \rceil$ colors to color the $k$-subsets of $\{1, \dots, n\}$ such that any $r$ of the $k$-subsets that have the same color are not pairwise disjoint. A generalization of this problem where one requires $s$-wise instead of pairwise intersections was considered by Sarkaria. He claimed a proof of a generalized Erdős--Kneser conjecture establishing a lower bound for the number of colors that reduces to Erdős' original conjecture for ${s = 2}$. Lange and Ziegler pointed out that his proof fail…
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…
California State University, Fullerton（富尔顿加州州立大学）；Fort Valley State University（福尔特谷州立大学）；University of Texas at Tyler（泰勒德州大学）；University of Kentucky（肯塔基大学）
We suggest a new method on coloring generalized Kneser graphs based on hypergraphs with high discrepancy and small number of edges. The main result is providing a proper coloring of K(n, n/2-t, s) in (4 + o(1))(s + t)^2 colors, which is produced by Hadamard matrices. Also, we show that for colorings by independent set of a natural type, this result is the best possible up to a multiplicative constant. Our method extends to Kneser hypergraphs as well.
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…
The orthogonality dimension of a graph $G=(V,E)$ over a field $\mathbb{F}$ is the smallest integer $t$ for which there exists an assignment of a vector $u_v \in \mathbb{F}^t$ with $\langle u_v,u_v \rangle \neq 0$ to every vertex $v \in V$, such that $\langle u_v, u_{v'} \rangle = 0$ whenever $v$ and $v'$ are adjacent vertices in $G$. The study of the orthogonality dimension of graphs is motivated by various applications in information theory and in theoretical computer science. The contribution of the present work is two-fold. First, we prove that there exists a constant $c$ such that for ever…
In this paper, we investigate circular chromatic number of Mycielski construction of graphs. It was shown in \cite{MR2279672} that $t^{\rm th}$ Mycielskian of the Kneser graph $KG(m,n)$ has the same circular chromatic number and chromatic number provided that $m+t$ is an even integer. We prove that if $m$ is large enough, then $χ(M^t(KG(m,n)))=χ_c(M^t(KG(m,n)))$ where $M^t$ is $t^{\rm th}$ Mycielskian. Also, we consider the generalized Kneser graph $KG(m,n,s)$ and show that there exists a threshold $m(n,s,t)$ such that $χ(M^t(KG(m,n,s)))=χ_c(M^t(KG(m,n,s)))$ for $m\geq m(n,s,t)$.
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 …
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>…
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…
This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable subgraph problem. We present various properties of $\vartheta_k(G)$, such as that the sequence $(\vartheta_k(G))_k$ is increasing and bounded from above by the order of the graph $G$. We study $\vartheta_k(G)$ when $G$ is the strong, disjunction or Cartesian product of two graphs. We provide closed form expressions for the generalized $\vartheta$-number on seve…
