A note on fall colorings of Kneser graphs

Achromatic numbers of Kneser graphs

Complete colorings have the property that any two color classes has at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers comes from complete colorings, with some additional requirement. In this paper, we estimate these numbers in the Kneser graph $K(n,k)$ for some values of $n$ and $k$. We give the exact value of the achromatic number of $K(n,2)$.

A Generalization of Kneser's Conjecture

We investigate some coloring properties of Kneser graphs. A star-free coloring is a proper coloring $c:V(G)\to \Bbb{N}$ 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 $c: V(G) \to \{1,2,..., t\}$ is called the star-free chromatic number of $G$ and denoted by $χ_s(G)$. In view of Tucker-Ky Fan's lemma, we show that for any Kneser graph ${\rm KG}(n,k)$ we have $χ_s({\rm KG}(n,k))\geq \max\{2χ({\rm KG}(n,k))-10, χ({\rm KG}(n,k))\}$ where $n\geq 2k \geq 4$. Moreover, we show that $…

Trivial colors in colorings of Kneser graphs

We show that any proper coloring of a Kneser graph $KG_{n,k}$ with $n-2k+2$ colors contains a trivial color (i.e., a color consisting of sets that all contain a fixed element), provided $n>(2+\varepsilon)k^2$, where $\varepsilon\to 0$ as $k\to \infty$. This bound is essentially tight. This is a consequence of a more general result on the minimum number of non-trivial colors needed to properly color $KG_{n,k}$.

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。

Hypergraphs with many Kneser colorings (Extended Version)

For fixed positive integers $r, k$ and $\ell$ with $1 \leq \ell < r$ and an $r$-uniform hypergraph $H$, let $κ(H, k,\ell)$ denote the number of $k$-colorings of the set of hyperedges of $H$ for which any two hyperedges in the same color class intersect in at least $\ell$ elements. Consider the function $\KC(n,r,k,\ell)=\max_{H\in{\mathcal H}_{n}} κ(H, k,\ell) $, where the maximum runs over the family ${\mathcal H}_n$ of all $r$-uniform hypergraphs on $n$ vertices. In this paper, we determine the asymptotic behavior of the function $\KC(n,r,k,\ell)$ for every fixed $r$, $k$ and $\ell$ and descr…

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," d…

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).

Coloring of the square of Kneser graph $K(2k+r,k)$

The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of an $n$ elements set, with two vertices adjacent if they 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 $K(n, k)$ is an interesting graph coloring problem, and is also related with intersecting family problem. The square of $K(2k, k)$ is a perfect matching and the square of $K(n, k)$ is the complete grap…

On b-continuity Of Kneser Graphs of type KG(2k+1,k)

In this paper, we will introduce an special kind of graph homomorphisms namely semi-locally-surjective graph homomorphisms and show some relations between semi-locally-surjective graph homomorphisms and colorful colorings of graphs and then we prove that for each natural number $k$, the Kneser graph $KG(2k+1,k)$ is $b$-continuous. Finally, we introduce some special conditions for graphs to be $b$-continuous.

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 …

Circular Coloring and Mycielski Construction

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 $m\geq m(n,s,t)$.

关于 Kneser 染色的两个问题

Two questions on Kneser colorings

本文研究 Kneser 图的两个染色问题：证明相交族并集大小的上界并在颜色数接近色数时说明其紧性，同时改进选择色数的下界至 $\frac{1}{25}n\log n$。

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…

Kneser 图中的 Ramsey 数

Ramsey Numbers in Kneser Graphs

本文定义并研究 Kneser 图中的 Ramsey 数，给出一般上下界，并推进了两个相关的 Ramsey 型问题。

Circular chromatic number of induced subgraphs of Kneser graphs

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 List Distinguishing Number of Kneser Graphs

A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is $k$-distinguishable. If for each $v\in V(G)$ we have a list $L(v)$ of colors, and we stipulate that the color assigned to vertex $v$ comes from its list $L(v)$ then $G$ is said to be $\mathcal{L}$-distinguishable where $\mathcal{L} =\{L(v)\}_{v\in V(G)}$. The list distinguishing number of a graph, denoted $D_l(G)$, is the minimum integer $k$ such that every co…

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。

Chromatic Number of Random Kneser Hypergraphs

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'…

On Fall Colorings of Graphs

A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. We denote ${\rm Fall}(G)$ the set of all positive integers $k$ for which $G$ has a fall $k$-coloring. In this paper, we study fall colorings of lexicographic product of graphs and categorical product of graphs and answer a question of \cite{dun} about fall colorings of categorical product of complete graphs. Then, we study fall colorings of union of graphs. Then, we prove that fall $k$-colorings of a graph can be reduced into proper $k$-colorings of gr…

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…

About Fall Colorings of Graphs

A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. In this paper, we answer some questions of \cite{dun} about some relations between fall colorings and some other types of graph colorings.

Fall-colorings and b-colorings of graph products

Given a proper coloring $f$ of a graph $G$, a b-vertex in $f$ is a vertex that is adjacent to every color class but its own. It is a b-coloring if every color class contains at least one b-vertex, and it is a fall-coloring if every vertex is a b-vertex. The b-chromatic number of $G$ is the maximum integer $b(G)$ for which $G$ has a b-coloring with $b(G)$ colors, while the fall-chromatic number and the fall-acromatic number of $G$ are, respectively, the minimum and maximum integers $f_1(G),f_2(G)$ for which $G$ has a fall-coloring. In this article, we explore the concepts of b-homomorphisms and…

Graph Homomorphisms via Vector Colorings

In this paper we study the existence of homomorphisms $G\to H$ using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number $t \ge 2$ for which there exists an assignment of unit vectors $i\mapsto p_i$ to its vertices such that $\langle p_i, p_j\rangle\le -1/(t-1),$ when $i\sim j$. Our approach allows to reprove, without using the Erdős-Ko-Rado Theorem, that for $n>2r$ the Kneser graph $K_{n:r}$ and the $q$-Kneser graph $qK_{n:r}$ are cores, and furthermore, that for $n/r = n'/r'$ there exists a homomorphism $K_{n:r}\to K_{n':…

Conflict-Free Coloring on Open Neighborhoods

In an undirected graph, a conflict-free coloring (with respect to open neighborhoods) is an assignment of colors to the vertices of the graph $G$ such that every vertex in $G$ has a uniquely colored vertex in its open neighborhood. The conflict-free coloring problem asks to find the smallest number of colors required for a conflict-free coloring. The conflict-free coloring problem is NP-complete. From results in Abel et. al. [SODA 2017], it can be inferred that every planar graph has a conflict-free coloring with at most nine colors. As the best known lower bound for planar graphs is four colo…

On Colorings of Graph Powers

In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose $(2t+1)$st power is bounded by a Kneser graph. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a $(2k+1)$-cycle exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$ is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. We also consider Nešetřil's Pentagon problem. This problem is…

Universal completability, least eigenvalue frameworks, and vector colorings

An embedding $i \mapsto p_i\in \mathbb{R}^d$ of the vertices of a graph $G$ is called universally completable if the following holds: For any other embedding $i\mapsto q_i~\in \mathbb{R}^{k}$ satisfying $q_i^T q_j = p_i^T p_j$ for $i = j$ and $i$ adjacent to $j$, there exists an isometry mapping the $q_i$'s to the $p_i$'s for all $ i\in V(G)$. The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjace…

Local chromatic number, Ky Fan's theorem, and circular colorings

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…

Kneser与Schrijver问题的固定参数算法

Fixed-Parameter Algorithms for the Kneser and Schrijver Problems

研究了Kneser和Schrijver图的计算问题，证明了寻找单色边存在运行时间为n^{O(1)}·k^{O(k)}的固定参数随机算法，并将其应用于Agreeable-Set问题获得多项式时间算法。

关于距离-衰减着色的一个注记

A Note on Distance-Fall Colorings

本文研究图的距离-衰减着色问题，证明了连通可3-着色图具有距离-2衰减3-着色，并由此证明了Beineke和Henning关于树中独立距离-d-支配集大小上界的猜想。

Number of Distinguishing Colorings and Partitions

A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a coloring. This paper is about counting non-equivalent distinguishing colorings of graphs with $k$ colors. A parameter, namely $Φ_k (G)$, which is the number of non-equivalent distinguishing colorings of a graph $G$ with at most $k$ colors, is shown here to have an application in calculating the distinguishing number of the lexicographic product and the $X$-join o…

Kneser问题的一个固定参数算法

A Fixed-Parameter Algorithm for the Kneser Problem

本文提出一个运行时间为$n^{O(1)} \cdot k^{O(k)}$的随机化算法，证明Kneser问题关于参数$k$是固定参数可解的，并将其应用于Agreeable-Set问题，得到随机化多项式时间算法。

曲面上的曲线着色

Coloring curves on surfaces

本研究探讨曲面曲线图色数，得出分离曲线色数增长规律、固定同调类曲线唯一着色性，给出曲线色数最佳界，还研究弧图变体及低复杂度曲面精确结果，关联Kneser图等领域。

Colorings of complements of line graphs

Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph $\operatorname{KG}(n,2)$ that have a chromatic number equal to its chromatic number, namely $n-2$. In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the …

无冲突着色：有界团宽度图与交图

Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs

该研究探讨图的无冲突着色问题，分析团宽度与最少着色数的关系，针对多种图类给出紧的上下界与判定复杂度，解决了区间图开邻域无冲突着色的公开问题。

The Erdős-Gyárfás function with respect to Gallai-colorings

For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to have a $(p, q)$-coloring. This function was introduced about 45 years ago, but was studied systematically by Erdős and Gyárfás in 1997, and is now known as the Erdős-Gyárfás function. In this paper, we study $f(n, p, q)$ with respect to Gallai-colorings, where a Gallai-coloring is an edge-coloring of $K_n$ without rainbow triangles. Combining the two concepts, we …

图的奇独立数，I：基础与经典类

The odd independence number of graphs, I: Foundations and classical classes

本文研究了图的奇独立数及其与强奇色数的关系，证明了无爪图的奇独立数等于其平方图的独立数，并探讨了多种图类的奇独立数特性。

Coloring Problems on Bipartite Graphs of Small Diameter

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, List $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving NP-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained through a proof that the Surjective $C_6$-Homomorphism problem is NP-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vika…

Approximate and exact results for the harmonious chromatic number

Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system. The results presented in this paper fall into two categories: in the first part of the paper we are concerne…

关于对边密度分布良好的图进行着色

On coloring graphs with well-distributed edge density

本文引入平均遗传图类，证明其色数的新上界并改进先前结果，同时证明3-着色问题在该类上仍为NP难，且可在多项式时间内判定该类性质。

Parameter Estimation for Undirected Graphical Models with Hard Constraints

The hardcore model on a graph $G$ with parameter $λ>0$ is a probability measure on the collection of all independent sets of $G$, that assigns to each independent set $I$ a probability proportional to $λ^{|I|}$. In this paper we consider the problem of estimating the parameter $λ$ given a single sample from the hardcore model on a graph $G$. To bypass the computational intractability of the maximum likelihood method, we use the maximum pseudo-likelihood (MPL) estimator, which for the hardcore model has a surprisingly simple closed form expression. We show that for any sequence of graphs $\{G_N…

Vertex isoperimetry and independent set stability for tensor powers of cliques

The tensor power of the clique on $t$ vertices (denoted by $K_t^n$) is the graph on vertex set $\{1, ..., t\}^n$ such that two vertices $x, y \in \{1, ..., t\}^n$ are connected if and only if $x_i \neq y_i$ for all $i \in \{1, ..., n\}$. Let the density of a subset $S$ of $K_t^n$ to be $μ(S) := \frac{|S|}{t^n}$, and let the vertex boundary of a set $S$ to be vertices which are incident to some vertex of $S$, perhaps including points of $S$. We investigate two similar problems on such graphs. First, we study the vertex isoperimetry problem. Given a density $ν\in [0, 1]$ what is the smallest pos…