Perfect colorings of $Z^2$: Nine colors

Local distributions for eigenfunctions and for perfect colorings of q-ary hypercube

Under study are the eigenfunctions and perfect colorings of the graph of $n$-dimensional $q$-ary Hamming space. We obtain the interdependence of local distributions of an eigenfunction in two orthogonal faces. We prove also an analogous result for perfect colorings.

The perfect 2-colorings of infinite circulant graphs with a continuous set of odd distances

A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours of any vertex depends only on the color of the vertex. We consider perfect colorings of Cayley graphs of the additive group of integers with generating set $\{1,-1,3,-3,5,-5,\dots, 2n-1,1-2n\}$ for a positive integer $n$. We enumerate perfect $2$-colorings of the graphs under consideration and state the conjecture generalizing the main result to an arbitrary…

关于核完美定向与来自错位排列指派的DP-着色的注记

A note on kernel-perfect orientations and DP-colorings from derangement assignments

该研究将Bondy-Bopanna-Siegel引理推广至错位排列指派下的DP-着色，并借此推导出无零符号列表着色的定向定理，拓展了相关图论着色结论的适用范围。

On perfect colorings of the halved 24-cube

A vertex 2-coloring of a graph is said to be perfect with parameters $(a_{ij})_{i,j=1}^k$ if for every $i,j\in\{1,...,k\}$ every vertex of color $i$ is adjacent with exactly $a_{ij}$ vertices of color $j$. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube $\{0,1\}^{24}$ with parameters $((20+c,256-c)(c,276-c))$ (i.e., with eigenvalue 20). We prove that such colorings exist for all $c$ from 1 to 128 except 1, 2, 4, 5, 7, 10, 13 and do not exist for $c=1, 2, 4, 5, 7$. Keywords: perfect coloring, equitable partition, hypercube, halved n-cube

从半总到公平总着色

From semi-total to equitable total colorings, and their induced partitions into perfect and total perfect codes

研究通过改进的Kempe算法获取对称立方图和笼图的公平总着色，探讨总着色猜想的可行性。

J(10,3)上的对称完美2-着色

Symmetric Perfect $2$-colorings on $J(10,3)$

本研究针对与第三大特征值及对称商矩阵相关的Johnson图J(n,3)完美2-着色，梳理n=6的已知构造，给出n=10时两种已知完美2-着色的新构造并证明其唯一性。

On perfect colorings of infinite multipath graphs

A coloring of vertices of a given graph is called perfect if the color structure of each ball of radius $1$ in the graph depends only on the color of the ball center. Let $n$ be a positive integer. We consider a lexicographic product of the infinite path graph and a graph $G$ that can be either the complete or empty graph on $n$ vertices. We give a complete description of perfect colorings with an arbitrary number of colors of such graph products.

多维卷积矩阵与子空间超图的完美着色应用于弯曲函数及相关设计

Multidimensional convolution matrices and perfect colorings of subspace hypergraphs applied for bent functions and related designs

本文通过多维卷积矩阵与子空间超图的完美着色理论，研究了与弯曲函数相关的组合设计及其应用。

Proof of the List Coloring Conjecture for line perfect multigraphs

We prove that for a line perfect multigraph the chromatic index is equal to the list chromatic index. This is a generalization of Galvin's result on bipartite multigraphs. Soon after the first version was submitted to arxiv, I found out that the same result has already been achieved in an older paper by Peterson and Woodall, by a similar but not entirely the same method.

On perfect 2-colorings of the q-ary n-cube

A coloring of the $q$-ary $n$-dimensional cube (hypercube) is called perfect if, for every $n$-tuple $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean-valued function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-dimensional face of the hypercube. Let $f=χ^S$ be a characteristic function of some subset $S$ of hypercube. In the present paper it is proven the inequality $ρ(S)q({\rm cor}(f)+1)\leq α(S)$, where ${\rm cor}(f)$ is the maximum degree of the correlation immunity of $f$, $α(…

平面半正镶嵌的完美精确着色

Perfect precise colorings of plane semiregular tilings

本文研究平面半正镶嵌的完美精确着色问题，为$k\leq 6$的某些$k$价半正镶嵌族构造了满足对称群置换与顶点邻接颜色约束的着色方案。

关于完美2-着色的极值性质

On extremal properties of perfect 2-colorings

研究完美2-着色与图划分的极值性质，指出多个谱界仅在完美2-着色上达到，并对充分正则图给出固定平均内部度下子集大小的新上界。

某些完美图和无圈图类别的支配染色

Dominated Coloring of Some Classes of Perfect and Unicyclic Graphs

本文刻画了色支配的完美图，并研究了几类无圈图的支配色数，为图染色理论提供了新结果。

On the connection between correlation-immune functions and perfect 2-colorings of the Boolean n-cube

A coloring of the Boolean $n$-cube is called perfect if, for every vertex $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-face of the Boolean $n$-cube. In the present paper it is proven that each Boolean function $χ^S$ ($S\subset E^n$) satisfies the inequality $${\rm nei}(S)+ 2({\rm cor}(S)+1)(1-ρ(S))\leq n,$$ where ${\rm cor}(S)$ is the maximum degree of the correlation immunity of $χ^S$, ${\rm nei} (S)= \frac{1}{|S|}\sum\lim…

Perfectly Colorable Graphs

We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $ω(H)$ many colors where $ω(H)$ is the clique number of $H$. A graph is perfectly colorable if it admits a perfect coloring. We show that the class of perfectly colorable graphs is exactly the class of perfect paw-free graphs. It follows that perfectly colorable graphs can be recognized and colored in linear time.

Five-coloring graphs on the Klein bottle

We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list.

最大基数的法定着色：一类完美树的线性时间算法

Quorum colorings of maximum cardinality in linear time for a subclass of perfect trees

本文针对一类完美树（同一深度顶点度数相同），设计了线性时间算法，同时求出其最大基数的法定着色（$\psi_q$-着色）及法定着色数，解决了Sahbi提出的开放问题。

无限方格图的完美着色：覆盖与孪生颜色

Perfect colorings of the infinite square grid: coverings and twin colors

本文研究无限方格图的完美着色与覆盖问题，证明所有此类着色要么对应图自同构群作用下的轨道划分，要么具有合并后仍保持完美着色的孪生颜色，并对孪生颜色情形进行了分类。

组合设计、差集与 bent 函数作为图和多重图的完美着色

Combinatorial designs, difference sets and bent functions as perfect colorings of graphs and multigraphs

本文建立独立集、超图横截集与完美着色的等价关系，并证明组合设计、$q$-设计、差集、Hadamard 矩阵及 bent 函数可表示为特定图或多重图的完美着色。

On forced periodicity of perfect colorings

We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which $d$-dimensional configurations and finite patterns are presented as formal power series and Laurent polynomials, respectively, in $d$ variables. We consider perfect colorings that are configurations such that the number of points of a given color in the neighborhood of any point depends only on the color of the point for some fixed relative neighborhood, and we show that by choosing the alphabet suitably any perfect coloring has a …

The threshold for the full perfect matching color profile in a random coloring of random graphs

Consider a graph $G$ with a coloring of its edge set $E(G)$ from a set $Q = \set{c_1,c_2, \ldots, c_q}$. Let $Q_i$ be the set of all edges colored with $c_i$. Recently, Frieze defined a notion of the perfect matching color profile denoted by $\mcp(G)$, which is the set of vectors $(m_1, m_2, \ldots, m_q) \in [n]^q$ such that there exists a perfect matching $M$ in $G$ with $|Q_i \cap M| = m_i$ for all $i$. Let $\a_1, \a_2, \ldots, \a_q$ be positive constants such that $\sum_{i=1}^q \a_i = 1$. Let $G$ be the random bipartite graph $G_{n,n,p}$. Suppose the edges of $G$ are independently colored w…

Perfect colorings of the 12-cube that attain the bound on correlation immunity

We construct perfect $2$-colorings of the $12$-hypercube that attain our recent bound on the dimension of arbitrary correlation immune functions. We prove that such colorings with parameters $(x,12-x,4+x,8-x$) exist if $x=0$, $2$, $3$ and do not exist if $x=1$. This is a translation into English of the original paper by D. G. Fon-Der-Flaass, "Perfect colorings of the $12$-cube that attain the bound on correlation immunity", published in Russian in Siberian Electronic Mathematical Reports, vol. 4 (2007) 292-295.

Online coloring a token graph

We study a combinatorial coloring game between two players, Spoiler and Algorithm, who alternate turns. First, Spoiler places a new token at a vertex in $G$, and Algorithm responds by assigning a color to the new token. Algorithm must ensure that tokens on the same or adjacent vertices receive distinct colors. Spoiler must ensure that the token graph (in which two tokens are adjacent if and only if their distance in $G$ is at most $1$) has chromatic number at most $w$. Algorithm wants to minimize the number of colors used, and Spoiler wants to force as many colors as possible. Let $f(w,G)$ be …

Algebraic properties of perfect structures

A perfect structure is a triple $(M,P,S)$ of matrices $M, P$ and $S$ of consistent sizes such that $MP = PS$. Perfect structures comprise similar matrices, eigenvectors, perfect colorings (equitable partitions) and graph coverings. In this paper we study general algebraic properties of perfect structures and characterize all perfect structures with identity or unity matrix $M$. Next, we consider a graph product generalizing most standard products (e.g. Cartesian, tensor, normal, lexicographic graph products). For this product we propose a construction of perfect structures and prove that it ca…

缺陷着色对子式（minor）而言是完美的

Defective coloring is perfect for minors

该研究证实了子式闭族缺陷色数等于标准构造下界的猜想，得到聚类色数与禁子式树深度的线性关系，改进了此前的指数界。

利用核完美化研究若干凯莱图的列表着色

List Coloring of some Cayley graphs using Kernel perfections

本文研究酉凯莱图及二面体群上凯莱图的列表色数，利用核完美化方法给出其可选性的精确值或界。

On weight distributions of perfect colorings and completely regular codes

A vertex coloring of a graph is called "perfect" if for any two colors $a$ and $b$, the number of the color-$b$ neighbors of a color-$a$ vertex $x$ does not depend on the choice of $x$, that is, depends only on $a$ and $b$ (the corresponding partition of the vertex set is known as "equitable"). A set of vertices is called "completely regular" if the coloring according to the distance from this set is perfect. By the "weight distribution" of some coloring with respect to some set we mean the information about the number of vertices of every color at every distance from the set. We study the wei…

关于n立方体的3次与(n-4)-相关免疫完美着色

On degree-$3$ and $(n-4)$-correlation-immune perfect colorings of $n$-cubes

该研究刻画了n维布尔超立方体Q_n的次数至多为3的完美着色及(n-4)-相关免疫完美着色，分别给出了其商矩阵特征值条件与对应布尔函数的次数、相关免疫性质等价描述。

基于路径计数的完美图着色多项式时间算法

A Polynomial-Time Algorithm for Coloring Perfect Graphs Based on Walk Counting

该研究提出基于图论操作的多项式时间算法，核心是通过迭代计算加权路径判定完美图是否含给定大小团，权重先统一初始化再依上次迭代路径计数更新，用于对完美图进行最优着色。

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely al…

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。

两类全新无限族完全图的列表边着色猜想

The List Edge-Coloring Conjecture for New Infinite Families

本文证明了两类全新无限族完全图的列表边着色猜想，通过互补模方法控制1-因子分解的Pfaffian符号，建立行列式-Pfaffian桥梁并发展带符号Burnside-Fourier方法，同时揭示了该方法在高层论证中的赋值障碍。

用更少颜色给t-完美图着色

Coloring t-perfect graphs with fewer colors

研究t-完美图色数，通过改进证明，利用线性界保证r-算术绳存在及稍弱算术绳概念，将t-完美图色数界从199053提高到186。

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…

完美图类上的缺陷着色

Defective Coloring on Classes of Perfect Graphs

研究缺陷着色问题在完美图类上的复杂性，证明其在分裂图和余图上为NP-困难，但在固定参数时对余图及平凡完美图有多项式算法，并给出余图的亚指数算法。

Graphes parfaits : structure et algorithmes

This work is the PhD thesis of Nicolas Trotignon, written in 2004 under the supervision of Frédéric Maffray. It is motivated by the desire for a better understanding of perfect graphs. The proof of the Claude Berge's perfect graph conjecture in 2002 by Chudnovsky, Robertson, Seymour and Thomas has shed a new light on this field of combinatorics. But some questions are still unsettled, particulary the existence of a combinatorial algorithm for the coloring of perfect graphs. An even pair of a graph is a pair of vertices such that every path joining them has even length. As proved by Fonlupt and…

Coloring graph classes with no induced fork via perfect divisibility

For a graph $G$, $χ(G)$ will denote its chromatic number, and $ω(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. An integer-valued function $f$ is called a $χ$-binding function for a hereditary class of graphs $\cal C$ if $χ(G) \leq f(ω(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a polynomial $χ$-binding function for t…

结构、完美可除性与($P_2\cup P_4, C_3$)-无环图的着色

Structure, Coloring, and Perfect Divisibility of $(P_2\cup P_4, C_3)$-Free Graphs

本文研究了($P_2\cup P_4, C_3$)-无环图的结构分解，证明了猜想成立，并确定了该类图的着色数。

关于 Doob 图和四元 Hamming 图中的多重完全码及其他一些完全正则码

On multifold perfect codes and some other completely regular codes in the Doob graphs and quaternary Hamming graphs

本文研究 Doob 图与四元 Hamming 图中的完全 2-着色，刻画多重 1-完全码参数，并在标准条件下证明大参数时完全 (b,c)-着色存在，同时构造覆盖半径为 2 的完全正则码。

列表染色与色可选择性的动态综述

List-Coloring and Chromatic-Choosability -- A Dynamic Survey

本文综述了列表染色和色可选择性的主要进展，重点介绍了已知等式成立的图类、存在非平凡间隙的图类以及证明这些结果的主要方法。

On completely regular codes of covering radius 1 in the halved hypercubes

We consider constructions of covering-radius-1 completely regular codes, or, equivalently, equitable 2-partitions (regular 2-partitions, perfect 2-colorings), of halved n-cubes. Keywords: completely regular code, equitable partition, regular partition, partition design, perfect coloring, halved hypercube.

完美可分割性与某些无叉图着色

Perfect divisibility and coloring of some fork-free graphs

本文证明(奇气球,叉)-自由图是完美可分割的，推广了Karthick等人的结果，并作为应用得出(叉,宝石)-自由或(叉,蝴蝶)-自由图的色数上界为binom(ω(G)+1,2)。

在 $\mathsf{GaTEx}$ 图上求解 NP-hard 问题：完美排序、团、着色和独立集的线性时间算法

Solving NP-hard Problems on \textsc{GaTEx} Graphs: Linear-Time Algorithms for Perfect Orderings, Cliques, Colorings, and Independent Sets

本文研究 $\mathsf{GaTEx}$ 图上的 NP-hard 问题，利用 galled-tree 结构作为指导，提出了在线性时间内求解最大团、最优顶点着色、完美排序和最大独立集的算法。

On Acyclic Edge-Coloring of Complete Bipartite Graphs

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $Δ= Δ(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = Δ+ 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus es…

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…

Finding unavoidable colorful patterns in multicolored graphs

We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $χ$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $χ$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $χ$ of $K_n$ contain a subgraph isomorphic…

The Strong Colors of Flowers - The Structure of Graphs with Chordal Squares

A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a more general coloring concept can be defined: The strong coloring of a graph. So a k-strong coloring is a coloring where two vertices may not have the same color if their distance to each other is at most k. The 2-strong coloring of the line graph is known as the strong edge coloring. Coloring the kth power G^k of a graph G is the same as finding a k-strong …

计数完美匹配的量子态系统

Quantum state systems that count perfect matchings

本文利用拓扑量子场论将基于 Penrose 公式的 $n$-色顶点多项式范畴化为大分次同调，并证明其滤过版本由面着色生成，在 $n=2$ 时对应完美匹配，从而赋予该多项式新的图不变量意义。

编码理论与密码学中的完美组合结构

Perfect Combinatorial Structures in Coding Theory and Cryptography

本书发展代数与组合方法，研究图的完美着色（等划分）及其与图论、编码理论、傅里叶分析等领域的联系，面向高年级学生和研究人员。

K₉₂的两个完美1-因子分解

Two Perfect One-Factorizations of $K_{92}$

针对100以内未知完美1-因子分解的阶数92，本文构造了K₉₂的两个非同构完美1-因子分解，推进了Kotzig关于完全图完美1-因子分解的猜想研究。

Symmetries and color symmetries of a family of tilings with a singular point

We obtain tilings with a singular point by applying conformal maps on regular tilings of the Euclidean plane, and determine its symmetries. The resulting tilings are then symmetrically colored by applying the same conformal maps on colorings of regular tilings arising from sublattice colorings of the centers of its tiles. In addition, we determine conditions so that the coloring of a tiling with singularity that is obtained in this manner is perfect.

An Exact Algorithm for the Generalized List $T$-Coloring Problem

The generalized list $T$-coloring is a common generalization of many graph coloring models, including classical coloring, $L(p,q)$-labeling, channel assignment and $T$-coloring. Every vertex from the input graph has a list of permitted labels. Moreover, every edge has a set of forbidden differences. We ask for such a labeling of vertices of the input graph with natural numbers, in which every vertex gets a label from its list of permitted labels and the difference of labels of the endpoints of each edge does not belong to the set of forbidden differences of this edge. In this paper we present …

通过列表着色限制临界图中的奇反洞

Restricting Odd Antiholes in Critical Graphs via List Coloring

本研究针对固定最大度的临界图，结合列表着色定理，探究奇反洞的结构，证明了临界图中奇反洞的长度上界，验证了极端情况的刻画，并构造反例说明下界可达，还提出了相关染色猜想。

Almost color-balanced perfect matchings in color-balanced complete graphs

For a graph $G$ and a not necessarily proper $k$-edge coloring $c:E(G)\to \{ 1,\ldots,k\}$, let $m_i(G)$ be the number of edges of $G$ of color $i$, and call $G$ {\it color-balanced} if $m_i(G)=m_j(G)$ for every two colors $i$ and $j$. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly $n$-edge colored complete bipartite graph $K_{n,n}$ has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whethe…