On the chromatic number of two generalized Kneser graphs

On the chromatic number of almost s-stable Kneser graphs

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…

The Chromatic Number of the $q$-Kneser Graph for $q \geq 5$

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 $q = 2$.

On the Chromatic Number of some generalized Kneser 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…

The chromatic number of almost stable Kneser hypergraphs

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…

Kneser 色函数区分树

The Kneser chromatic function distinguishes trees

本文证明 Kneser 色函数 $X_{K_{\mathbb{N},2}}$ 是树的完全不变量，从而推进了 Stanley 关于色对称函数完全区分树的猜想。

A counterexample to a conjecture on the chromatic number of r-stable Kneser hypergraphs

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 $$χ\left(\textrm{KG}^r\left(\mathcal{F}_{r-stab}\right)\right)\geq\left\lceil\frac{cd_r\left(\mathcal{F}\right)}{r-1}\right\rceil.$$

Colorful hypergraphs in Kneser hypergraphs

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).

On Chromatic Number of Kneser Hypergraphs

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…

On the chromatic number of generalized Kneser hypergraphs

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.

Coloring Properties of Categorical Product of General Kneser Hypergraphs

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…

On the locating chromatic number of Kneser graphs

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 …

On the chromatic number of almost stable general Kneser hypergraphs

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…

Chromatic numbers of Kneser-type graphs

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…

高色数情形的匹配 Kneser 图猜想

The Matching Kneser Graph Conjecture For High Chromatic Numbers

本文证明了在满足特定参数条件下，存在连通图使得其团数、色数、$rK_2$-染色数及边数减去 $rK_2$-极值边数满足给定等式，从而支持高色数情形的匹配 Kneser 图猜想。

The chromatic number of a two families of generalized Kneser graphs related to finite generalized quadrangles and finite projective 3-spaces

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…

G-稳定Kneser图的色数

The chromatic number of $G$-stable Kneser graphs

本文定义G-稳定Kneser图，研究k-紧图的判定问题，证明k≥3时k-紧图的必要条件并猜想其充分，还利用G为不交圈并时的色数下界证明了特定4-正则图存在大小为k的独立集。

类 Sierpiński 图与 Kneser 图的单射着色

Injective colorings of Sierpiński-like graphs and Kneser graphs

本文建立单射色数与色数、边色数的关系，确定 Sierpiński 图及根积的单射色数，并刻画 Sierpiński 图与 Kneser 图的完美单射可着色性。

三角剖分的Lovász-Kneser定理

A Lovász-Kneser theorem for triangulations

本文证明了凸n边形的三角剖分的Kneser图的色数为n-2，扩展了Lovász-Kneser定理在几何三角剖分领域的应用。

Hedetniemi's conjecture for Kneser hypergraphs

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…

A note on b-coloring of Kneser graphs

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].

Colorful Subhypergraphs in Uniform Hypergraphs

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…

On the circular chromatic number of a subgraph of the Kneser graph

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…

交错奇环与类Kneser图的定向

Alternating odd cycles and orientations of Kneser-like graphs

本文研究特定图的定向问题，聚焦拓扑χ-染色图，探讨其是否存在避免交错奇环的定向或使所有最短奇环为交错奇环的定向，并针对Kneser图等多个图族给出结果。

Improved bounds on the chromatic numbers of the square of Kneser graphs

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…

Graphs whose Kronecker covers are bipartite Kneser graphs

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.

A new lower bound for the chromatic number of general Kneser hypergraphs

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…

Counterexamples to a conjecture on matching Kneser 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…

高效的谱界：汉明图、约翰逊图和克奈塞尔图幂的色数界

Efficient spectral bounds on the chromatic number of Hamming, Johnson, and Kneser graph powers

本文提出利用超几何正交多项式高效计算汉明图、约翰逊图和克奈塞尔图幂的谱界，从而在多项式时间内得出色数的下界。

Lower bounds for the chromatic number of certain Kneser-type hypergraphs

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…

PG$(3,q)$ 的 chambers 上的 Kneser 图的最大余团与色数

Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG$(3,q)$

本文研究有限射影三维空间 chambers 的 Kneser 图，刻画其极大独立集大小与结构，并证明色数为 $q^2+q$。

关于投影平面室的克罗内克图着色数的注记

A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets

本文通过简单证明展示了在投影平面的 incidence 图中，X 和 Y 的补集之间存在完美匹配，从而将投影平面室的克罗内克图着色数问题与 incidence 图的 incidence-free 数问题联系起来。

稳定 Kneser 超图的色数：验证新族猜想

On the Chromatic Number of Stable Kneser Hypergraphs: Verifying the Conjecture for New Families

本研究针对稳定 Kneser 超图色数猜想，在 s>r≥3 情形下，通过分析向量稳定 Kneser 图，首次对无穷多参数给出严格理论证明，覆盖 r=4, s=6 及 r=4, k=2, s=5 等新族。

New construction of graphs with high chromatic number and small clique

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.

Hadwiger's conjecture for the complements of Kneser graphs

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.

Chromatic Number Via Turan Number

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 …

Haggkvist-Hell Graphs: A class of Kneser-colorable graphs

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…

Sharp bounds for the chromatic number of random Kneser graphs

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 $p$. …

Kneser 图中的全奇分拆

Totally odd subdivisions in Kneser graphs

本文加强了 Kneser 图中关于 Odd Hadwiger Conjecture 的已有结果，证明了存在包含任意大完全全奇分拆的 $t$-色 Kneser 图，且所有 Kneser 图均含 $K_{χ(G)}$ 的全奇分拆，并扩展至全奇浸入研究。

On random subgraphs of Kneser and Schrijver graphs

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_…

通过共识划分研究Kneser超图的色数

The Chromatic Number of Kneser Hypergraphs via Consensus Division

本文利用共识划分定理为Kneser超图色数下界提供新证明，并证明对每个素数p，Kneser^p问题可高效归约到弱近似的共识划分问题且属于PPA-p，从而给出其复杂性限制。

广义Kneser图和广义Johnson图的零阻塞数

The zero blocking numbers of generalized Kneser graphs and generalized Johnson graphs

本文推广了Kneser图和Johnson图零阻塞数的已知结果至广义版本。

Random Kneser graphs and hypergraphs

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 …

3 - 稳定克内泽尔图的色数

The chromatic number of 3-stable Kneser graphs

研究3 - 稳定克内泽尔图的色数，证明了梅尼耶猜想在\(s = 3\)且\(n\)足够大或\(k = s = 3\)时成立，通过证明\(s\) - 稳定集的希尔顿 - 米尔纳定理版本及采用拓扑方法实现。

On some topological and combinatorial lower bounds on chromatic number of Kneser type hyper graphs

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>…

关于广义Kneser超图染色

On generalized Kneser hypergraph colorings

在Ziegler（2002）中，第二作者给出了超图$\KG{r}{\pmb s}{\calS}$的色数的一个下界，“广义的$r$ -均匀Kneser交交重数超图$\pmb s$ .“它推广了Kriz（1992/2000）对没有交重数的${\pmb s}=(1,...,1)$的下界和Sarkaria（1990）对$\calS=\tbinom{[n]}k$的下界。在这里，我们讨论的微妙之处和困难，出现交重数$s_i>1$：1。在存在交重数的情况下，有两种不同版本的“Kneser超图”，这取决于是否允许超图的边是多集而不是集。我们证明了超图的这两个概念的色数是有本质区别的。Sarkaria（1990）和Ziegler（2002）的下界仅适用于多集版本。2.在Sarkaria和Ziegler的证明中，对素数$r$的简化仅在交重数数严格小于$r$的最大素因子时才起作用。目前我们还没有有效的证据证明其他情况下的下界结果。我们还证明了所有没有多重集边的一致超图都可以表示为广义Kneser超图。

Chromatic number, Clique number, and Lovász's bound: In a comparison

In the way of proving Kneser's conjecture, László Lovász settled out a new lower bound for the chromatic number. He showed that if neighborhood complex $\mathcal{N}(G)$ of a graph $G$ is topologically $k$-connected, then its chromatic number is at least $k+3$. Then he completed his proof by showing that this bound is tight for the Kneser graph. However, there are some graphs where this bound is not useful at all, even comparing by the obvious bound; the clique number. For instance, if a graph contains no complete bipartite graph $\mathcal{K}_{l, m}$, then its Lovász's bound is at most $l+m-1$.…

Intersection patterns of finite sets and of convex sets

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…

圆着色与Mycielski构造

Circular Coloring and Mycielski Construction

本文研究了图的Mycielski结构的圆色数。在\cite{MR2279672}中证明了：当$m+t$为偶数时，Kneser图$KG(m,n)$的$t^{\rm th}$ Mycielskian具有相同的圈色数和色数.我们证明了如果$m$足够大，则有$χ(M^t(KG(m,n)))=χ_c(M^t(KG(m,n)))$，其中$M^t$是$t^{\rm th}$ Mycielskian的.此外，我们考虑了广义Kneser图$KG(m,n,s)$，并证明了存在一个阈值$m(n,s,t)$，使得$χ(M^t(KG(m,n,s)))=χ_c(M^t(KG(m,n,s)))$对于$m\geq m(n,s,t)$。

A Characterization of Distance Matrices of Positive Weighted Kneser Graphs and Generalized Petersen Graphs

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.

全Kneser图的自同态

Endomorphisms of Total Kneser Graphs

受曲线图色数研究启发，证明全Kneser图的所有自同态均为自同构，并确定其自同构群为两置换群之积。

Critical subgraphs of Schrijver graphs for the fractional chromatic number

Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.

Erdös-Ko-Rado定理的推广

A Generalization of the Erdös-Ko-Rado Theorem

本文研究了文献[8]中定义的局部Kneser图的一些性质。在这方面，作为Erd ${\rm \ddot{o}}$ s-Ko-Rado定理的推广，我们刻画了局部Kneser图的最大独立集.其次，我们给出了它们的色数的一个上界。

On the neighborhood complex of $\vec{s}$-stable Kneser graphs

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…

Treewidth of the generalized Kneser graphs

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 $K(n,k,k-1)$ us…