Precoloring Extension. IV. General Bounds and List Colorings

在图的笛卡尔积中扩展有限大小的边着色

Extending partial edge-colorings of bounded size in Cartesian products of graphs

本文研究图笛卡尔积中的边预着色扩展问题，提出猜想并证明在特定条件下，边预着色可扩展为适当着色。

可平面图的列表预着色扩张

List precoloring extension in planar graphs

一个著名的结果是，不仅每个可平面图都可以用五种颜色适当地着色，而且无论如何将五种颜色的任意调色板分配给顶点，都可以为每个顶点从相应的调色板中选择一种颜色，从而得到的着色是适当的。这个结果被称为可平面图的5-可选性。艾伯森问是否可以通过预着色一些顶点是在一个足够大的距离，除了在一个图中扩展的Lassen定理。在这里，除其他外，我们回答的问题的情况下，当图不包含短周期分离precolored顶点，当有一个“宽”斯坦纳树包含所有的precolored顶点。

Extending partial edge colorings of cartesian products of graphs

We consider the problem of extending partial edge colorings of cartesian products of graphs. More specifically, we suggest the following Evans-type conjecture: If $G$ is a graph where every precoloring of at most $k$ precolored edges can be extended to a proper $χ'(G)$-edge coloring, then every precoloring of at most $k+1$ edges of $G \square K_2$ is extendable to a proper $(χ'(G) +1)$-edge coloring of $G \square K_2$. In this paper we verify that this conjecture holds for trees, complete and complete bipartite graphs, as well as for graphs with small maximum degree. We also prove versions of …

推广预着色判别群作用

Extending Precolorings to Distinguish Group Actions

给定一个作用于集合$X$的群$Γ$，如果唯一固定$ϕ$的$γ\in Γ$是身份动作，则$X$的$k$着色$ϕ:X\to\{1,\dots,k\}$相对于$Γ$是有区别的。动作$Γ$的区别数，表示为$D_Γ(X)$，则是最小的正整数$k$，使得存在$X$相对于$Γ$的区别$k$ -着色。这个概念已经在许多设置中进行了研究，但到目前为止，最大的工作机构一直关注于寻找图PDMATHAMT的自同构群在其顶点集上的作用的区别数，这被称为PDMATHAMT的区别数。一个群体行动的区别数是衡量“打破”该行动产生的所有排列的难度的一个尺度。在本文中，我们的目的是进一步区分具有相同的可区分数的群作用的弹性。特别是，我们引入了一个预着色扩展框架来解决这个问题。集合$S \subseteq X$是$Γ$的固定集合，如果对于每个非单位元素$γ\in Γ$，存在元素$s \in S$使得$γ(s) \neq s$。可区分的扩张数$\operatorname{ext}_D(X,Γ;k)$是最小数$m$，使得对于所有具有$|W| \geq m$的固定集$W \subseteq X$，每个$k$ -着色$c : X \setminus W \to [k]$都可以被扩张为可区分$X$的$k$ -着色.本文证明了$\operatorname{ext}_D(\mathbb{R},\operatorname{Aut}(\…

分数预着色的扩张表现出不连续性

Extensions of Fractional Precolorings show Discontinuous Behavior

本文研究了如下问题：给定一个实数k和整数d，使一个分数k-可着色图的任意两两距离至少为d的点的分数（k+ n）-预着色可以推广为整个图的分数（k+ n）-着色的最小的n是多少？对于k=2和k \ge 3以及任何d，我们确定了k \in（2，3）（如果d=4）和k \in [2.5，3）（如果d=6）的精确值，并给出了k \in（2，3）（如果d= 5，7）和k \in（2，2.5）（如果d=6）的上界。令人惊讶的是，作为k的函数的k对于所有这些d值都是不连续的。

Precoloring extension of Vizing's Theorem for multigraphs

Let $G$ be a graph with maximum degree $Δ(G)$ and maximum multiplicity $μ(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $Δ(G)+μ(G)$. The distance between two edges $e$ and $f$ in $G$ is the length of a shortest path connecting an endvertex of $e$ and an endvertex of $f$. A distance-$t$ matching is a set of edges having pairwise distance at least $t$. Edwards et al. proposed the following conjecture: For any graph $G$, using the palette $\{1, \dots, Δ(G)+μ(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge co…

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…

曲面上具有可变列表大小的列表着色图

List-coloring graphs on surfaces with varying list-sizes

设$G$是一个嵌入在曲面$S_\varepsilon$上的图，其Euler亏格为$\varepsilon > 0$，$P\subseteq V(G)$是一组顶点，它们之间的距离至少为4.假设$G$的所有顶点都有$H(\varepsilon)$ -列表，并且$P$的顶点都是预着色的，其中$H(\varepsilon)=\Big\lfloor\frac{7 + \sqrt{24\varepsilon + 1}}{2}\Big\rfloor$是Heawood数。我们证明了$P$的着色扩展到$G$的列表着色，并且距离界为4是最佳可能的。我们的结果回答了Albertson关于将可平面图中一组相互远离的顶点的预着色扩展为图的5-列表着色的类似问题，并将Albertson和Hutchinson的结果推广到曲面上的列表着色扩展.

Linear Hypergraph List Edge Coloring - Generalizations of the EFL Conjecture to List Coloring

Motivated by the Erdős-Faber-Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and Vizing's theorem for graphs. For example, we conjecture that in a linear hypergraph of rank 3, the list edge chromatic number is at most 2 times the maximum degree plus 1. We show that for sufficiently large fixed rank and sufficiently large degree, the conjectures are true.

List Supermodular Coloring with Shorter Lists

In 1995, Galvin proved that a bipartite graph $G$ admits a list edge coloring if every edge is assigned a color list of length $Δ(G)$, the maximum degree of the graph. This result was improved by Borodin, Kostochka and Woodall, who proved that $G$ still admits a list edge coloring if every edge $e=st$ is assigned a list of $\max\{d_{G}(s), d_{G}(t)\}$ colors. Recently, Iwata and Yokoi provided the list supermodular coloring theorem, that extends Galvin's result to the setting of Schrijver's supermodular coloring. This paper provides a common generalization of these two extensions of Galvin's r…

Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8

We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring. Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin.

On differences between DP-coloring and list coloring

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle. Many known upper bounds for the list-chromatic number extend to the DP-chromatic number, but not all of them do. In this note we describe some unusual properties of DP-coloring that set it aside from list coloring. In particular, we give an example of a planar bipartite graph with DP-chromatic number $4$ and prove that the edge-DP-chromatic number of a $d$-regular graph with $d\geq 2$ is always at least $d+1$.

最大度至多为5的三角剖分的列表边着色

List-Edge-Coloring Triangulations with Maximum Degree at most 5

该研究证明了最大度不超过5的三角剖分满足列表边着色猜想，推进了图论中列表边着色相关问题的研究进展。

Colorings v.s. list colorings of uniform hypergraphs

Let $r$ be an integer with $r\ge 2$ and $G$ be a connected $r$-uniform hypergraph with $m$ edges. By refining the broken cycle theorem for hypergraphs, we show that if $k>\frac{m-1}{\ln(1+\sqrt{2})}\approx 1.135 (m-1)$ then the $k$-list assignment of $G$ admitting the fewest colorings is the constant list assignment. This extends the previous results of Donner, Thomassen and the current authors for graphs.

$2$-distance, injective, and exact square list-coloring of planar graphs with maximum degree 4

In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graph…

Note on list star edge-coloring of subcubic graphs

{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvořák et al. (Star chromatic index, J. Graph Theory 72 (2013), 313--326).

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。

Generalization of some results on list coloring and DP-coloring

In this work, we introduce DPG-coloring using the concepts of DP-coloring and variable degeneracy to modify the proofs on the following papers: (i) DP-3-coloring of planar graphs without $4$, $9$-cycles and cycles of two lengths from $\{6, 7, 8\}$ (R. Liu, S. Loeb, M. Rolek, Y. Yin, G. Yu, Graphs and Combinatorics 35(3) (2019) 695-705), (ii) Every planar graph without $i$-cycles adjacent simultaneously to $j$-cycles and $k$-cycles is DP-$4$-colorable when $\{i, j, k\}=\{3, 4, 5\}$ (P. Sittitrai, K. Nakprasit, arXiv:1801.06760(2019) preprint), (iii) Every planar graph is $5$-choosable (C. Thoma…

图的平方的着色、列表着色和涂色（及其他相关问题）

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

本文综述了图的平方的着色、列表着色和涂色问题，特别是强边着色，主要关注平面图和其他稀疏图类。

Partial DP-Coloring

In 1980, Albertson and Berman introduced partial coloring. In 2000, Albertson, Grossman, and Haas introduced partial list coloring. Here, we initiate the study of partial coloring for an insightful generalization of list coloring introduced in 2015 by Dvořák and Postle, DP-coloring (or correspondence coloring). We consider the DP-coloring analogue of the Partial List Coloring Conjecture, which generalizes a natural bound for partial coloring. We show that while this partial DP-coloring conjecture does not hold, several results on partial list coloring can be extended to partial DP-coloring. We…

曲面上图的Heawood列表染色的一个变形

A variation on Heawood list-coloring for graphs on surfaces

我们证明了一个变化的Heawood列表着色的表面上的图形，仿照Mesquassen的平面5-列表着色定理。对于φ>0，将希伍德数定义为H（φ）=Floor（（7+Sqrt[24* φ +1]）/2）。本文证明了：除n =3外，嵌入到Euler亏格n>0的曲面上的每个图F都是列表着色的，当F的顶点有（H（n）-2）-列表，其它顶点都有H（n）-列表时，除非F的顶点上的诱导子图包含H（n）-1顶点上的完全图.

Analogue of DP-coloring on variable degeneracy and its applications on list vertex-arboricity and DP-coloring

In \cite{listnoC3adjC4}), Borodin and Ivanova proved that every planar graph without $4$-cycles adjacent to $3$-cycle is list vertex-$2$-aborable. In fact, they proved a more general result. Inspired by these results and DP-coloring which becomes a widely studied topic, we introduce a generalization on variable degeneracy including list vertex arboricity. We use this notion to extend a general result by Borodin and Ivanova. Not only that this theorem implies results about planar graphs without $4$-cycles adjacent to $3$-cycle by Borodin and Ivanova, it implies many other results including a re…

List coloring with requests

Let G be a graph with a list assignment L. Suppose a preferred color is given for some of the vertices; how many of these preferences can be respected when L-coloring G? We explore several natural questions arising in this context, and propose directions for further research.

固定颜色数的在线列表着色

Online List Colorings with the Fixed Number of Colors

在线列表着色是图论中一个广泛研究的课题。一个图$G$是2-可画的，如果我们总是有一个策略来完成$G$的在线列表着色，其中每个顶点都有一个大小为2的颜色列表。本文主要研究了在线列表着色游戏，其中颜色的数量是事先已知的。我们说$G$是$[2,t]$ -paintable的，如果我们总是有一个策略来完成$G$的在线列表着色中的着色，其中我们提前知道恰好有$t$颜色，并且每个顶点都有一个大小为2的颜色列表。令$M(G)$表示其中$G$不是$[2,t]$可着色的最大$t$，并且$m(G)$表示其中$G$不是$[2,t]$可着色的最小$t \geq 2$。我们证明了如果$G$不是2-可画的，那么$2 \leq m(G) \leq 4,$和$n \leq M(G) \leq 2n-3.$。此外，我们分别用$m(G)\in \{2,3,4\}$和$M(G) \in \{n, n+1, 2n-3\},$刻画了$G$。

DP-3-coloring of planar graphs without $4,9$-cycles and two cycles from $\{5,6,7,8\}$

A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring. We note that list-coloring results do not always extend to DP-coloring results. Our main result in this paper is to prove that every planar graph without cycles of length $\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8…

On List Equitable Total Colorings of the Generalized Theta Graph

In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A $k$-assignment, $L$, for a graph $G$ assigns a list, $L(v)$, of $k$ available colors to each $v \in V(G)$, and an equitable $L$-coloring of $G$ is a proper coloring, $f$, of $G$ such that $f(v) \in L(v)$ for each $v \in V(G)$ and each color class of $f$ has size at most $\lceil |V(G)|/k \rceil$. In 2018, Kaul, Mudrock, and Pelsmajer subsequently introduced the List Equitable Total Coloring Conjecture which states that if $T$ is a total graph of some simple graph, then $T$ is…

List strong edge coloring of some classes of graphs

A {\em strong edge coloring} of a graph is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} of a graph is the minimum number of colors needed to obtain a strong edge coloring. In an analogous way, we can define the list version of strong edge coloring and list version of strong chromatic index. In this paper, we prove that if $G$ is a graph with maximum degree at most four and maximum average degree less than $3$, then the list strong chromatic index is at most $3Δ+ 1$, where $Δ$ is the maximum degree of $G$. In addition, we prove that …

列表着色中的一些概念

Some concepts in list coloring

本文研究了唯一列表可着色图.一个图G被称为是唯一k-列表可着色的，如果它允许一个k-列表分配，从该分配G有一个唯一的列表着色。G不是唯一k-列表可染的最小k称为G的M-数。本文证明了任何无三角形且色数为色数k+1的图都是唯一k-列表可染的。给出了图的M-数的一个界，并利用这个界证明了每个可平面图的M-数至多为4。在图中引入了列表临界性，刻画了所有3-列表临界图.证明了每个$χ_\ell$ -临界图都是$χ'$ -临界图，并证明了这个猜想与著名的列表着色猜想是等价的.

图的列表多数边着色

List majority edge-colorings of graphs

本文将有限图的多数边着色理论推广到无限图，在列表设定下证明了无悬挂边的无限图可用大小为4的列表实现多数边着色，并由此证实了线图的不友好划分猜想。

拟阵的列表着色与基交换性质

List coloring of matroids and base exchange properties

一个拟阵的着色是给它的基集的元素分配颜色。我们只限于适当的着色--那些相同颜色的元素形成一个独立集的着色。Seymour证明了一个$k$ -可着色拟阵也可以从任何大小为$k$的列表中着色。我们将这个定理推广到列表仍然具有固定大小但不一定相等的情况。对于任何固定大小的列表分配$\ell$，我们证明，如果一个拟阵是着色从一个特定的列表大小$\ell$，那么它是着色从任何列表大小$\ell$。这给出了一个显式的充要条件，使一个拟阵从任何固定大小的列表中可列表着色。作为一个应用，我们展示了如何使用我们的条件，以获得几个基地交换性质。

Five-list-coloring graphs on surfaces III. One list of size one and one list of size two

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $ϕ$ of $G$ such that $ϕ(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every $v\in V(C)-\{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)-V(C)$, then $G$ has an $L$-coloring. What happens when $v_1$ and $v_2$ are not adjacent? Then an $L$-coloring need not exist, but in the first paper of this series we have shown that it exists if $|L(v_1)|,|L(v_2)|\…

Total Equitable List Coloring

An equitable coloring is a proper coloring of a graph such that the sizes of the color classes differ by at most one. A graph $G$ is equitably $k$-colorable if there exists an equitable coloring of $G$ which uses $k$ colors, each one appearing on either $\lfloor |V(G)|/k \rfloor$ or $\lceil |V(G)|/k \rceil$ vertices of $G$. In 1994, Fu conjectured that for any simple graph $G$, the total graph of $G$, $T(G)$, is equitably $k$-colorable whenever $k \geq \max\{χ(T(G)), Δ(G)+2\}$ where $χ(T(G))$ is the chromatic number of the total graph of $G$ and $Δ(G)$ is the maximum degree of $G$. We investig…

Exponentially Many 4-List-Colorings of Triangle-Free Graphs on Surfaces

Thomassen proved that every planar graph $G$ on $n$ vertices has at least $2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for $G$ and $G$ has girth at least five. Postle and Thomas proved that if $G$ is a graph on $n$ vertices embedded on a surface $Σ$ of genus $g$, then there exist constants $ε,c_g > 0$ such that if $G$ has an $L$-coloring, then $G$ has at least $c_g2^{εn}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ or if $L$ is a 3-list-assignment for $G$ and $G$ has girth…

The list-coloring function of signed graphs

It is known that, for any $k$-list assignment $L$ of a graph $G$, the number of $L$-list colorings of $G$ is at least the number of the proper $k$-colorings of $G$ when $k>(m-1)/\ln(1+\sqrt{2})$. In this paper, we extend the Whitney's broken cycle theorem to $L$-colorings of signed graphs, by which we show that if $k> \binom{m}{3}+\binom{m}{4}+m-1$ then, for any $k$-assignment $L$, the number of $L$-colorings of a signed graph $Σ$ with $m$ edges is at least the number of the proper $k$-colorings of $Σ$. Further, if $L$ is $0$-free (resp., $0$-included) and $k$ is even (resp., odd), then the lo…

不含4-圈和5-圈的可平面图的正方形列表染色

List-coloring the Squares of Planar Graphs without 4-Cycles and 5-Cycles

设$G$是一个不含4-圈和5-圈的平面图，它具有最大度$Δ\ge 32$ .我们证明了$χ_{\ell}(G^2)\le Δ+3$ .对于任意大的最大度$Δ$，存在围长为6的可平面图$G_Δ$，且$χ(G_Δ^2)=Δ+2$ .因此，我们的界限在1以内是最优的。此外，我们的界限来自着色greetings在一个良好的秩序，所以立即扩展到在线列表着色的界限。此外，我们还证明了$L(p,q)$ -标记的界。具体地，$λ_{2,1}(G)\le Δ+8$，更一般地，$λ_{p,q}(G)\le (2q-1)Δ+6p-2q-2$，对于正整数$p$和$q$，具有$p\ge q$。同样，这些边界来自贪婪着色，所以它们立即扩展到这个问题的列表着色和在线列表着色变体。

部分表着色的一个下界

A Lower Bound for Partial List Colorings

设G是一个n点图，其表色数为$χ_\ell$ .假设G的每个顶点都被赋予一个t颜色的列表。Albertson、Grossman和Haas推测至少$t n / {χ_\ell}$顶点可以从这些列表中着色。我们证明了可着色顶点数的一个下界。作为一个推论，我们证明了至少有6/7的限制数可以着色。

Extracting list colorings from large independent sets

We take an application of the Kernel Lemma by Kostochka and Yancey to its logical conclusion. The consequence is a sort of magical way to draw conclusions about list coloring (and online list coloring) just from the existence of an independent set incident to many edges. We use this to prove an Ore-degree version of Brooks' Theorem for online list-coloring. The Ore-degree of an edge $xy$ in a graph $G$ is $θ(xy) = d_G(x) + d_G(y)$. The Ore-degree of $G$ is $θ(G) = \max_{xy\in E(G)}θ(xy)$. We show that every graph with $θ\ge18$ and $ω\le\fracθ{2}$ is online $\left\lfloor \fracθ{2}\right\rfloor …

关于部分表着色的一个注记

A Note on Partial List Colorings

设$G$是一个点数为$n$且列色数为$χ_\ell(G)=χ_\ell$的单图.假设为$0\leq t\leq χ_\ell$和$G$的每个顶点分配了一个$t$颜色列表。Albertson、Grossman和Haas [1]指出，至少$G$的$\frac{tn}{χ_\ell}$顶点可以从这些列表中着色。在本文中，我们发现了一些新的结果，有助于我们表明，猜想是正确的，至少有一半的号码集$\{1,2,...,χ_\ell(G)-1\}$的部分列表着色。此外，我们还引入了一个新的相关猜想，最后给出了关于这个猜想的一些结果。

Restricted extension of sparse partial edge colorings of complete graphs

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

拟阵的在线列表着色

On-line list coloring of matroids

若相同颜色的元素构成一个独立集，则称拟阵的染色是真的。Seymour的一个定理断言，一个k-可着色拟阵也可以从任何大小为k的列表中着色。本文将这一定理推广到在线情形。我们证明，着色的拟阵从大小为k的列表是可能的，即使在列表中的颜色的外观恢复颜色的颜色由对手，而我们的工作是分配一个颜色后，立即恢复。我们还证明了一个更一般的加权版本，我们的结果与不同大小的列表。由此得到了一般情况下拟阵列表可着色的一个简单的充要条件。我们使用的主要工具是多基交换性质，我们给出了一个简单的证明。

和表染色，和选择数与sc-贪婪图

Sum list coloring, the sum choice number, and sc-greedy graphs

设G =（V，E）是一个图，f是一个函数，它给G的顶点分配列表大小.称G是f-可选的，如果对于G的每个顶点的色表的赋值，如果其色表的大小与f一致，则存在G从这些色表中的正常染色。和选择数是G的所有可选择函数f上f的列表大小之和的最小值。图的和选择数总是至多为和|V| + |E|。当G的和选择数等于这个上界时，称G是sc-贪婪的.本文确定了所有五点图的和选择数，证明了圈树是sc-贪婪的，并给出了和表染色的一些新的一般性结果。

5-列表着色球面6正则三角剖分的线性时间算法

$5$-list-coloring toroidal $6$-regular triangulations in linear time

本文提出了一种在线性时间内对球面6正则三角剖分进行5-列表着色的算法，并证明此类图不满足3-可选择性。

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 DP-Coloring of Digraphs

DP-coloring is a relatively new coloring concept by Dvořák and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\{(v,c) ~|~ v \in V(G), c \in L(v)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle…

Proportional Choosability: A New List Analogue of Equitable Coloring

In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A $k$-assignment $L$ for a graph $G$ specifies a list $L(v)$ of $k$ available colors for each vertex $v$ of $G$. An $L$-coloring assigns a color to each vertex $v$ from its list $L(v)$. For each color $c$, let $η(c)$ be the number of vertices $v$ whose list $L(v)$ contains $c$. A proportional $L$-coloring of $G$ is a proper $L$-coloring in which each color $c \in …

Restricted extension of sparse partial edge colorings of hypercubes

We consider the following type of question: Given a partial proper $d$-edge coloring of the $d$-dimensional hypercube $Q_d$, and lists of allowed colors for the non-colored edges of $Q_d$,can we extend the partial coloring to a proper $d$-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.

稀疏图的公平列表着色

Equitable list coloring of sparse graphs

研究证明稀疏图在特定条件下的公平3-和4-可选性，引入强公平列表着色概念并给出上界。

稀疏超图的列表着色数

The list coloring number of uncrowded hypergraphs

研究证明，对于固定整数r≥2和任意ε>0，足够大的稀疏(r+1)超图在最大度Δ下，列表着色数上限为(1+ε)(rΔ/logΔ)^{1/r}。通过半随机列表着色nibble方法，结合主动边着色约束和二项式度数界控制，最终完成着色。

List strong edge-coloring of graphs with maximum degree 4

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by $χ_{s}'(G)$ which is the minimum number of colors that allow a strong edge-coloring of $G$. Erdős and Nešetřil conjectured in 1985 that the upper bound of $χ_{s}'(G)$ is $\frac{5}{4}Δ^{2}$ when $Δ$ is even and $\frac{1}{4}(5Δ^{2}-2Δ+1)$ when $Δ$ is odd, where $Δ$ is the maximum degree of $G$. The conjecture is proved right when $Δ\leq3$. The best known upper bound for $Δ=4$ is 22 due to Cranston previously. In this paper we…

具有半小列表规模的二部图着色

Coloring bipartite graphs with semi-small list size

本研究针对二部图非对称列表着色问题，通过固定完全二部图一部列表规模研究渐近行为，发现决定可选性的关键不变量，结合超图独立集问题改进列表规模为2时的界，并提出统一已有三个猜想的一般二部图猜想。

具有不同列表大小的列表着色，完全二部图的情况

List colorings with distinct list sizes, the case of complete bipartite graphs

设$f:V \rightarrow \mathbb{N}$是图$G=(V,E)$的顶点集上的函数。图$G$是{ \em $f$ -choosable}，如果对于具有由$f$指定的列表大小的列表的每个集合，存在使用来自列表的颜色的适当着色。总和选择数$χ_{sc}(G)$是所有函数$f$上的$\sum f(v)$的最小值，使得$G$是$f$可选择的。已知（Alon 1993，2000），如果$G$具有平均度$d$，则通常的选择数$χ_\ell(G)$至少为$Ω(\log d)$，因此它们同时增长。本文证明了当最小度$χ_{sc}(G)/|V(G)|$.我们的主要工具是给出不平衡完全二部图$K_{a,q}$的和选择数的紧估计。

区间图的列表可区别数等于可区别数

The List Distinguishing Number Equals the Distinguishing Number for Interval Graphs

图$G$的\textit{distinguishing coloring}是顶点的着色，使得$G$的每个非平凡自同构将某个顶点映射到具有不同颜色的顶点。$G$的\textit{distinguishing number}是最小$k$，使得$G$具有区别着色，其中每个顶点被分配来自$\{1,\ldots,k\}$的颜色。到$G$的\textit{list assignment}是颜色列表到$G$的顶点的分配$L=\{L(v)\}_{v\in V(G)}$。$G$的\textit{distinguishing $L$-coloring}是$G$的区别着色，其中每个顶点$v$的颜色来自$L(v)$。$G$的{ \it列表区分数}是最小的$k$，使得对$G$的每个列表分配，其中所有$v\in V(G)$的$|L(v)|=k$产生$G$的区分$L$ -着色。证明了如果$G$是区间图，则它的可区别数与列可区别数相等.

List-three-coloring graphs with no induced $P_6+rP_3$

For an integer $r$, the graph $P_6+rP_3$ has $r+1$ components, one of which is a path on $6$ vertices, and each of the others is a path on $3$ vertices. In this paper we provide a polynomial-time algorithm to test if a graph with no induced subgraph isomorphic to $P_6+rP_3$ is three-colorable. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of $\{1,2,3\}$.

有界树宽图的符号列表边着色

Signed list edge coloring in graphs of bounded treewidth

本文研究符号图的列表边着色，将经典列表边着色与符号边着色推广，证明树宽为3的符号图及树宽4且Δ≥10的符号图满足 Vizing 猜想的符号类似。