Some Ramsey theorems for finite $n$-colorable and $n$-chromatic graphs
L. Nguyen Van Thé
给定一个固定的整数$n$，我们证明了所有有限序$n$ -可着色图，有限$n$ -可着色图，有限序$n$ -色图和有限$n$ -色图类的Ramsey型定理.
Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.
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We give an upper bound on the list chromatic number of a 2-colorable hypergraph which generalizes the bound of Schauz on $k$-partite $k$-uniform hypergraphs. It makes sense for sparse hypergraphs: in particular we show that a $k$-uniform $k$-regular hypergraph has the list chromatic number 2 for $k \geq 4$. Also we obtain both lower and upper bound on the list chromatic number of a complete $s$-uniform 2-colorable hypergraph in the vein of Erd{\H o}s--Rubin--Taylor theorem.
We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.
Large chromatic number and Ramsey graphs
设Q（n，c）表示n-顶点图的色数为c时所能拥有的最小团大小。使用Ramsey图，我们给出了一个精确的，虽然隐含的，公式的情况下c至少是（n+3）/2。
Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we determine the Gallai-Ramsey numbers for connected graphs with five vertices of chromatic number four.
We prove for k at most 10, that every graph of chromatic number k with a unique k-coloring admits a clique minor of order k.
Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log t})$-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no $K_t$ minor, we make the first improvement on this bound by showing that every graph with no odd $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$.
Equitable chromatic threshold of complete multipartite graphs
如果图的色类的大小至多相差1，则图的正常点着色是公平的。图$G$的均匀色数记为$χ_=(G)$，是使$G$均匀$k$可染的最小$k$.图$G$的公平色阈值，记为$χ_=^*(G)$，是使$G$对$k\ge t$公平$k$可着色的最小$t$.给出了计算任意完全多部图的公平色阈值的公式和线性时间算法。
The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J. Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star $6$-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether $χ'_s(G)\le3…
Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable graphs have been completely proved to convince all1-5, but the proofs are tremendously difficult for over the 5-colorable graph6,7. Although the development of graph theory inspires scientists to understand graph coloring deeply, it is still an open problem for over 7-colorable graphs6,7. Therefore, we put forward a brand new chromatic graph configuration and …
A graph $G$ is Ramsey for a graph $H$ if every colouring of the edges of $G$ in two colours contains a monochromatic copy of $H$. Two graphs $H_1$ and $H_2$ are Ramsey equivalent if any graph $G$ is Ramsey for $H_1$ if and only if it is Ramsey for $H_2$. A graph parameter $s$ is Ramsey distinguishing if $s(H_1)\neq s(H_2)$ implies that $H_1$ and $H_2$ are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multi-colour case and use a similar idea to find another graph parameter which is Ramsey distinguishing.
Examples of topologically highly chromatic graphs with locally small chromatic number
Kierstead，Szemerédi和Trotter证明了一个图有至多个$\lfloor r/(2n)\rfloor^n$顶点，使得其中每个半径为$r$的球都是$c$ -可着色的，该图应该有至多色数$n(c-1)+1$。我们表明，这一估计是尖锐的$r$。也就是说，对于每一个$n$，$r$，和$c$，我们构造一个图$G$包含$O((2rc)^{n-1}c)$顶点，使得$χ(G)\geq n(c-1)+1$，虽然每个球的半径$r$在$G$是$c$ -着色。其核心思想是构造一个图，其邻域复形同伦等价于两个给定图的邻域复形之并。
The following relaxation of the classical problem of determining Ramsey number of a fixed graph has first been proposed by Erdos, Hajnal and Rado over 50 years ago. Given a graph $G$ and an integer $t \geq 2$ determine the minimum number $N$ such that in any $t$-coloured complete graph on $N$ vertices there is a copy of $G$ using only edges of some $t-1$ colours. We determine the answer precisely when $G$ is a path.
A hole is an induced cycle of length at least 4. A graph is called a pentagraph if it has no cycles of length 3 or 4 and has no holes of odd length at least 7, and is called a heptagraph if it has no cycles of length less than 7 and has no holes of odd length at least 9. Let $ł\ge 2$ be an integer. The current authors proved that a graph is 4- colorable if it has no cycles of length less than $2ł+1$ and has no holes of odd length at least $2ł+3$. Confirming a conjecture of Plummer and Zha, Chudnovsky and Seymour proved that every pentagraph is 3-colorable. Following their idea, we show that ev…
On the chromatic number of random regular graphs
设G（n，d）是n阶随机d-正则图.对于任何超过某个常数k_0的整数k，我们确定一个数d_{k-col}，使得G（n，d）是k-可着色的，如果d < d_{k-col} and non-k-colorable w.h.p. if d>d_{k-col}。
On the strong chromatic number of random graphs
设G是一个n阶图，k是n的整除整数.称G是强k-可着色的，如果对V（G）的每个不相交集V_1 \cup. \cup V_r，都是大小为k的，存在G的一个正常顶点k-染色，每个颜色在每个V_i中只出现一次。在k不整除n的情况下，如果由k个\lceil n/k个\rceil - n孤立点相加得到的图是强k-可着色的，则G被定义为强k-可着色的.图G的强色数是使G是强k-可染的最小k。本文研究了随机图G（n，p）中该参数的性质。在稠密情形下，当p >> n^{-1/3}时，我们证明了强色数是a.s.集中在一个值Δ +1上，其中Δ是图的最大度。对于稀疏随机图，我们也得到了一些较弱的结果。
An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the vertex set. The $t$-color Ramsey number $R_t(G)$ of an ordered graph $G$ is the minimum number of vertices of an ordered complete graph such that every edge-coloring from a set of $t$ colors contains a monochromatic copy of $G$ such that the copy of $G$ preserves the original ordering on $G$. An ordered graph is $k$-ichromatic if it has interval chromatic number …
A sub-exponential transition of the chromatic generalized Ramsey numbers
一个简单的图积型构造表明，对于所有自然数$r \ge q$，存在完全图在$2^r$顶点上的$r$颜色的边染色，其中由任意$q$颜色类的并集组成的图具有色数$2^q$ .本文证明了对于每个固定的自然数$q$，如果存在完全图在$n$顶点上的边染色，且由任意$q$色类的并集组成的图的色数至多为$2^q -1 $，则$n$在$r$中一定是次指数的.这回答了康伦、福克斯、李和苏达科夫的一个问题。
Replacing the host K_n by n-chromatic graphs in Ramsey-type results
将拉姆齐理论中的两个著名结果从$K_n$推广到任意的$n$ -色图.首先是一个注意到Erd \H OS和雷达指出，在每2着色的边缘$K_n$有一个单色树上$n$顶点。第二个是Cockayne和Lorimer的定理，指出对于满足$n_1=\max\{n_1,n_2,\dots,n_t\}$和$n=n_1+1+\sum_{i=1}^t (n_i-1)$的正整数，以下成立。在$K_n$的边与颜色$1,2\dots,t$的每个着色中，对于某些$i\in \{1,2,\dots,t\}$，存在大小$n_i$的单色匹配。
We show that every planar graph $G$ has a 2-fold 9-coloring. In particular, this implies that $G$ has fractional chromatic number at most $\frac92$. This is the first proof (independent of the 4 Color Theorem) that there exists a constant $k<5$ such that every planar $G$ has fractional chromatic number at most $k$.
Degrees in oriented hypergraphs and Ramsey p-chromatic number
由于$D(k,m)$在估计有向图的最大有向割和有向图的最小有向割覆盖中所起的作用，最近几篇论文研究了一类有向图族$D(k,m)$，使得对于每个顶点$v \in V(G)$，要么（出度）$°^+(v) \le k$，要么（入度）$°^-(v) \le m$.利用图的$d$ -退化度的概念，得到了图族$D(k,m)$中图的色数的一些结果.在本文中，我们考虑了一个深远的推广家庭$D(k,m)$，在一个补充的形式，到$r$-一致超图的背景下，使用Hakimi的定理推广到$r$ -一致超图，并显示出一些紧密的联系与著名的Ramsey数超图。
The chromatic number for properly colouring the facets of a combinatorial simple $n$-polytope $P^n$ that is the orbit space of a quasitoric manifold satisfies the inequality $n\leq P^n\leq 2^n-1$. The inequality is sharp for $n=2$ but not for $n=3$ due to the Four Color theorem. In this note, we construct a simple 4-polytope admitting a characteristic map whose chromatic number equals $15$ and deduce that the predicted upper bound is attained for $n=4$. Analogues results are verified for the case of oriented small covers in dimensions $4$ and $5$.
Uniquely 2-List Colorable Graphs
一个图称为唯一列表可着色的，如果它允许一个列表分配，导致一个唯一的列表着色。我们研究唯一列表可着色图的颜色使用的数量上的限制。在这种方法中，我们推广了一个定理，唯一的2-列表着色图的特点。本文引入了图的唯一色数表，并对它作了一个猜想，它是著名的Brooks定理的推广。
Wegner conjectured in 1977 that the square of every planar graph with maximum degree at most $3$ is $7$-colorable. We prove this conjecture using the discharging method and computational techniques to verify reducible configurations.
A note on the list chromatic number of two matroids
本文研究两个拟阵的公共独立集的列表着色，构造例子表明公共色数与公共列表色数可不等，并证明两个强基序拟阵若各自2-可着色则其交集2-列表可着色。
We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t.
Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we study Gallai-Ramsey numbers for graphs with chromatic number three such as $\widehat{K}_m$ for $m\ge2$, where $\widehat{K}_m$ is a kipas with $m+1$ vertices obtained from the join of $K_1$ and $P_m$, and a class of graphs with five vertices, denoted by $\mathscr{H}$. We first study the general lower bound of s…
Number of vertices in graphs with locally small chromatic number and large chromatic number
讨论了一个具有大色数的图的最小顶点数，使得其中每个固定半径的球都具有小的色数。本文证明了：对于每个在$\sim((n+rc)/(c+rc))^{r+1}$顶点上的图$G$，使得每个半径为$r$的球都是$c$ -可着色的，我们有$χ(G)\leq n$ .
We show that for every graph $G$ and every graph $H$ obtained by subdividing each edge of $G$ at least $O(\log |V(G)|)$, $H$ is nonrepetitively 3-colorable. In fact, we show that $O(\log π'(G))$ subdivisions per edge are enough, where $π'(G)$ is the nonrepetitive chromatic index of $G$. This answers a question of Wood and improves a similar result of Pezarski and Zmarz that stated the existence of at least one 3-colorable division with a linear number of subdivision vertices per edge.
It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has at least one color that appears an odd number of times in its neighborhood. The smallest number of colors that admits an odd coloring of a graph $ G $ is denoted by $χ_{o}(G)$. In this paper, we prove that if $G$ is tortoidal, then $χ_{o}\left({G}\right)\le9$; Note that $K_7$ is a toroidal graph, the upper bound is no less than $7$.
The chromatic gap and its extremes
{ \em色间隙}是图的色数与团数之差。在这里，我们研究$\gap(n)$，最大色间隙的图上的$n$顶点。极值图能被探索吗？虽然与色隙有关的计算问题是无望的，但拉姆齐理论和匹配理论之间的相互作用导致了一个简单且（几乎）精确的$\gap(n)$公式。
The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved …
A coloring is called $s$-wide if no walk of length $2s-1$ connects vertices of the same color. A graph is $s$-widely colorable with $t$ colors if and only if it admits a homomorphism into a universal graph $W(s,t)$. Tardif observed that the value of the $r^{\rm th}$ multichromatic number $χ_r(W(s,t))$ of these graphs is at least $t+2(r-1)$ and equality holds for $r=s=2$. He asked whether there is equality also for $r=s=3$. We show that $χ_s(W(s,t))=t+2(s-1)$ for all $s$ thereby answering Tardif's question. We observe that for large $r$ (with respect to $s$ and $t$ fixed) we cannot have equalit…
Hadwiger's conjecture states that every $K_t$-minor free graph is $(t-1)$-colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd $K_t$-minor is $(t-1)$-colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form $Ct$ for some constant $C>0$ exists. We show that if every graph without a $K_t$-minor is $f(t)$-colorable, then every graph without an odd $K_t$-minor is $2f(t)$-colorable. Using this, th…
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。
Planar Graphs with Ore-degree at Most seven is strongly $13$-edge-colorable
本文证明Ore度至多7的平面图强色指数不超过13，解决Chen等人的猜想，采用放电法与组合零点定理。
A generalization of Ramsey theory for stars and one matching
广义拉姆齐理论中最近提出的一个问题是，对于固定的正整数$s\leq t$，在$K_n$的每一条边用$t$颜色着色时，至少有多少个顶点可以被不超过$s$的单色成员族$\cal F$的顶点覆盖.这与Chung和Liu提出的{ $d$ -色Ramsey数}有关。在本文中，我们首先计算这些数字的恒星推广了著名的结果伯尔和罗伯茨。然后我们推广了Cockayne和Lorimer的一个结果，计算了恒星和一个匹配的$d$ -色Ramsey数.
Claw-free cubic graphs are $(1, 1, 2, 2)$-colorable
本文证明了每个无爪三次图都存在 $(1,1,2,2)$-着色，从而验证了次三次图细分图的packing chromatic number至多为 $5$ 的猜想在无爪三次图情况下成立。
Chromatic Ramsey number of acyclic hypergraphs
设$T$是一个无圈的$r$ -一致超图，且$r\ge 2$ .我们定义（$t$ -color）色Ramsey数$χ(T,t)$为最小的$m$，具有以下性质：如果任何$m$ -色$r$ -均匀超图的边以任何方式用$t$颜色着色，则存在$T$的单色副本。我们观察到$χ(T,t)$是定义良好的，并且$$\left\lceil {R^r(T,t)-1\over r-1}\right \rceil +1 \le χ(T,t)\le |E(T)|^t+1$$其中$R^r(T,t)$是$H$的$t$ -颜色Ramsey数。当T是匹配或星形时，我们给出了$χ(T,t)$的线性上界，证明了对于$r\ge 2, k\ge 1, t\ge 1$，$χ(M_k^r,t)\le (t-1)(k-1)+2k$和$χ(S_k^r,t)\le t(k-1)+2$，其中$M_k^r$和$S_k^r$分别是$r$ -一致匹配和具有$k$边的星形.改进了$3$ -一致超图的一般界.证明了$χ(M_k^3,2)=2k$，推广了Alon-Frankl-Lovász定理的一个特例.我们还证明了$χ(S_2^3,t)\le t+1$，这是尖锐的$t=2,3$。这是一个更一般的结果的推论。我们定义$H^{[1]}$为$H$的1-交图，其顶点表示超边，其边表示超边在一个顶点上的交点。证明了对任意$3$ -一致超图$H$（假设$χ(…
Haggkvist-Hell Graphs: A class of Kneser-colorable graphs
对于正整数n和r，我们定义Haggkvist-Hell图H_{n:r}为顶点为有序对（h，T）的图，其中T是[n]的r-子集，h是[n]中不在T中的元素。顶点（h_x，T_x）和（h_y，T_y）相邻当且仅当h_x \in T_y，h_y \in T_x，且T_x和T_y不相交.这些无三角形弧传递图是Kneser图思想的推广，并且从Haggkvist-Hell图H_{n:r}到相应的Kneser图K_{n:r}存在一个自然同态. Haggkvist和Hell介绍了这些图的r=3的情况，证明了一个三次图允许同态到H_{22:3}当且仅当它是无三角形的。Gallucio、Hell和Nesetril也考虑了r=3的情况，证明了H_{n:3}可以有任意大的色数。本文给出了所有Haggkvist-Hell图的直径、围长和奇围长的精确值，并给出了独立性、色数和分数色数的界.进一步，我们将Gallucio等人的结果推广到任意固定的r-\ge 2，并确定了H_{n:r}的全自同构群，它同构于n元上的对称群.
We apply Ramsey theoretic tools to show that there is a family of graphs which have tree-chromatic number at most~$2$ while the path-chromatic number is unbounded. This resolves a problem posed by Seymour.
Chromatic Ramsey numbers and two-color Turán densities
研究图的双色图兰密度，证明对任意k色图G，色数拉姆齐数R_χ(G)可取Ω(k)个不同值，揭示了双色图兰密度的可能取值。
$K_3$-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
图$G$的$K_3$ -WORM着色是指对图的顶点进行颜色分配，使得$G$的每个$K_3$ -子图的顶点恰好得到两种颜色。我们研究图$G$，承认至少有一个这样的着色。我们反驳了戈达德等人的一个猜想[大会编号，219（2014）161--173]，他询问是否每个这样的图都有两种颜色的$K_3$ -WORM着色。事实上，对于每一个整数$k\ge 3$，存在一个$K_3$ -WORM可着色图，其中的最小颜色数正好是$k$。还存在$K_3$可着色图，其具有带有两种颜色的$K_3$着色以及带有$k$颜色但不带有任何$3,\dots,k-1$颜色的着色.我们还证明了它是NP-困难的，以确定的最小数量的颜色和NP-完全决定$k$ -可着色性的每一个$k \ge 2$（仍然棘手，即使是最大程度9，如果$k=3$的图）。另一方面，我们证明了$d$ -退化图与小$d$的积极结果，也包括平面图。此外，我们指出了一个基本的联系，混合超图的染色理论。最后，我们列出了许多悬而未决的问题。
Star Coloring on Some Subclasses of Chordal Graphs
研究弦图子类的星着色，给出星3-可着色弦图的结构和禁止诱导子图刻画，以及星4-和星5-可着色分裂图的禁止诱导子图刻画，并设计线性时间认证识别算法。
A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in $K_4$ . Let $H$ be the complement of a cycle on 7 vertices. Chudnovsky et al [6] proved that every (odd hole, $K_4$)-free graph is 4-colorable and is 3-colorable if it does not has $H$ as an induced subgraph. In this paper, we use the proving technique of Chudnovsky et al to generalize this conclusion to (odd hole, full house)-free graphs, and prove that for (odd hole, full house)-free graph $G$, $χ(G)\le ω(G)+1$, and the equ…
In order to solve a question on list coloring of planar graphs, Dvořák and Postle introduced the concept of so called DP-coloring, thereby extending the concept of list-coloring. DP-coloring was anaylized in detail by Bernshteyn, Kostochka, and Pron for graphs and multigraphs; they characterized DP-degree colorable multigraphs and deduced a Brooks' type result from this. The characterization of the corresponding 'bad' covers was later given by Kim and Ozeki. In this paper, the concept of DP-colorings is extended to hypergraphs having multiple (hyper-)edges. We characterize the DP-degree colora…
Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $Δ$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with chromatic index $Δ(G)+k$ ($k\geq 1$) can be decomposed into a maximum $Δ(G)$-edge-colorable subgraph (not necessarily class I) and a $k$-edge-colorable subgraph. In this paper, we first generalize their result to multigraphs and show that every multigraph $G$ with multiplicity $μ$ can be decomposed into a maximum $Δ(G)$-edge-colorable subgraph and a subgraph with ma…
An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $χ_o(G)$, is the minimum number of colors that admits an odd coloring of $G$. Petruševski and Škrekovski in 2021 introduced this notion and proved that if $G$ is planar, then $χ_o(G)\le9$ and conjectured that $χ_o(G)\le5$. More recently, Petr and Portier improved $9$ to $8$. A graph is $1$-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showe…
Size of edge-critical uniquely 3-colorable planar graphs
一个图$G$是\emph{uniquely k-colorable}，如果$G$的色数是$k$，并且$G$只有一个$k$ -染色直到颜色的置换.唯一$k$可着色图$G$是边临界图，如果$G-e$对任何边$e\in E(G)$都不是唯一$k$可着色图. Mel'nikov和Steinberg [L. S.梅尔尼科夫河Steinberg，One counterexample for two aesthetures on three coloring，Discrete Math.20（1977）203-206]要求找到一个边临界3-可着色平面图的边数的精确上界。本文给出了边临界唯一3-可着色可平面图的一些性质，并证明了如果$G$是这样的具有$n(\geq6)$顶点的图，则$\frac{8}{3}n-\frac{17}{3}$改进了Matsumoto [N. Matsumoto，边临界唯一3-可着色可平面图的尺寸，Electron. J. Combin。20（3）（2013）$\#$ P49]。此外，我们还找到了一些边临界3-可着色的可平面图，它们的顶点为$n(=10,12, 14)$，边为$\frac{5}{2}n-7$.
Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. It is known that for a fixed $n$ one has \[ c_n r^n < m(n,r) < C_n r^n. \] We prove that for any fixed $n$ the sequence $a_r := m(n,r)/r^n$ has a limit, which was conjectured by Alon. We also prove the list colorings analogue of this statement.
On r-equitable chromatic threshold of Kronecker products of complete graphs
一个图$G$是$r$-公平$k$-可着色的，如果它的顶点集可以被划分成$k$独立集，其中任何两个在大小上最多相差$r$。图$G$的$r$-公平色阈值，由$χ_{r=}^*(G)$表示，是最小$k$，使得$G$对于所有$k'\ge k$都是$r$-公平地$k'$-可着色的。设$G\times H$表示图$G$和$H$的Kronecker积.本文完全确定了一般$m,n$和$r$的$χ_{r=}^*(K_m\times K_n)$的精确值。因此，我们证明了，对于$r\ge 2$，如果$n\ge \frac{1}{r-1}(m+r)(m+2r-1)$，那么$K_m\times K_n$和它的生成超图$K_{m(n)}$具有相同的$r$-公平可着色性，特别是$χ_{r=}^*(K_m\times K_n)=χ_{r=}^*(K_{m(n)})$，其中$K_{m(n)}$是在每个部分中具有$n$顶点的完全$m$-部图。
The $s$-chromatic Ramsey number for stars
本文定义了$s$-色拉姆齐数和星临界$s$-色拉姆齐数，并完全确定了星图$K_{1,m}$的这两个值，部分确定了多星图的$s$-色拉姆齐数。
Two results on the digraph chromatic number
已知（Bollobás（1978）; Kostochka和Mazurova（1977））存在最大度$Δ$和任意大围长的图，其色数至少为$c Δ/ \log Δ$。我们证明了一个类似的结果，其中有向图$D$的色数定义为最小整数$k$，使得$V(D)$可以划分为$k$非循环集，围长是相应的无向图中最短圈的长度。与Erdos（1962）的一个旧结果相同，本文还证明了存在具有任意大色数的有向图，其中每个大的顶点子集都是2-可着色的。
In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Jo…
Four plane unit vectors generate a $3$-colorable graph
本文证明任意四个平面单位向量生成的含正负生成元的Cayley图均为3可着色，并将其推广，确定了关系群秩不超2时四元及其逆生成的阿贝尔Cayley图的色数。
We study the computational efficiency of approaches, based on Hilbert's Nullstellensatz, which use systems of linear equations for detecting non-colorability of graphs having large girth and chromatic number. We show that for every non-$k$-colorable graph with $n$ vertices and girth $g>4k$, the algorithm is required to solve systems of size at least $n^{Ω(g)}$ in order to detect its non-$k$-colorability.
Conventional Ramsey-theoretic investigations for edge-colourings of complete graphs are framed around avoidance of certain configurations. Motivated by considerations arising in the field of Qualitative Reasoning, we explore edge colourings that in addition to forbidding certain triangle configurations also require others to be present. These conditions have natural combinatorial interest in their own right, but also correspond to qualitative representability of certain nonassociative relation algebras, which we will call chromatic.
Some Results on incidence coloring, star arboricity and domination number
建立了连接关联色数、星荫度和控制数三个孤立图不变量的两个不等式。由此，我们得到了所有图的关联色数的上界和下界。利用这些界，我们进一步降低了可平面图的关联色数的上界，并证明了阶数不可被4整除的三次图是不可4-关联着色的。确定了图的笛卡儿积、并、并的关联色数。
Chromatic thresholds for pairs of graphs
研究图对的色阈值问题，确定当图中无红H1或蓝H2副本时，色数有界的最小度密度。3色图对的色阈值取五个值之一，由普通色阈值和嵌入C5型Ramsey配置决定。
On the smallest eigenvalues of $3$-colorable graphs
本文证明3-可着色图的最小特征值在$(-\infty,-\lambda^*)$内稠密，并据此说明球面二距离集研究中无法借助带符号图色数进一步改进禁用子图方法。
Equitable coloring of Kronecker products of complete multipartite graphs and complete graphs
如果图的色类的大小相差不超过1，则图的正常点着色是公平的。图$G$的均匀色数记为$χ_=(G)$，是使$G$均匀$k$可染的最小$k$.图$G$的公平色阈值，记为$χ_=^*(G)$，是使$G$对$k \ge t$公平$k$可着色的最小$t$.本文给出了$\sum_{i = 1}^r m_i \leq n$的$χ_=(K_{m_1,..., m_r} \times K_n)$和$χ_=^*(K_{m_1,..., m_r} \times K_n)$的精确值。
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