Generalized list colouring of graphs

List $4$-colouring of planar graphs

This paper proves the following result: If $G$ is a planar graph and $L$ is a $4$-list assignment of $G$ such that $|L(x) \cap L(y)| \le 2$ for every edge $xy$, then $G$ is $L$-colourable. This answers a question asked by Kratochvíl, Tuza and Voigt in [Journal of Graph Theory, 27(1):43--49, 1998].

局部平面图的无环列表着色

Acyclic List Colouring Locally Planar Graphs

该研究聚焦局部平面图的无环列表着色问题，证明局部平面图是无环9-列表可着色的，填补了此前该类图无环列表着色无固定色数上界的研究空白。

平面图列表着色3-路径可扩展性的多项式方法

The polynomial method for 3-path extendability of list colourings of planar graphs

本文用图多项式重述Thomassen的3-可扩展性定理，导出其Alon-Tarsi等价形式，扩展了平面5-可选性定理。

On Galvin orientations of line graphs and list-edge-colouring

The notion of a Galvin orientation of a line graph is introduced, generalizing the idea used by Galvin in his landmark proof of the list-edge-colouring conjecture for bipartite graphs. If L(G) has a proper Galvin orientation with respect to k, then it immediately implies that G is k-list-edge-colourable, but the converse is not true. The stronger property is studied in graphs of the form `bipartite plus an edge', the Petersen graph, cliques, and simple graphs without odd cycles of length 5 or longer.

部分列表染色猜想是错误的

The Partial List Colouring Conjecture is False

本文构造了一个14顶点、列表色数为3的图，其存在2-列表分配使最多9个顶点可正确着色，从而否证了部分列表染色猜想，反例由ChatGPT 6 Astra Ultra发现并验证。

具有局部列表大小的图的边着色

Edge-colouring graphs with local list sizes

本研究针对图的边着色问题，将Kahn关于列表着色猜想渐近成立的定理推广至局部情形，同时给出线性$k$-均匀超图、对应着色等相关推广的局部版本，并通过加权结论统一推导所有结果。

List Colouring Squares of Planar Graphs

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $Δ\ge8$ is at most $\bigl\lfloor\frac32Δ\bigr\rfloor+1$. We show that it is at most $\frac32 Δ(1+o(1))$ (where the $o(1)$ is as $Δ\to+\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs.

Partial list colouring of certain graphs

Let $G$ be a graph on $n$ vertices and let $\mathcal{L}_k$ be an arbitrary function that assigns each vertex in $G$ a list of $k$ colours. Then $G$ is $\mathcal{L}_k$-list colourable if there exists a proper colouring of the vertices of $G$ such that every vertex is coloured with a colour from its own list. We say $G$ is $k$-choosable if for every such function $\mathcal{L}_k$, $G$ is $\mathcal{L}_k$-list colourable. The minimum $k$ such that $G$ is $k$-choosable is called the list chromatic number of $G$ and is denoted by $χ_L(G)$. Let $χ_L(G) = s$ and let $t$ be a positive integer less than …

List colouring of graphs with at most $\big(2-o(1)\big)χ$ vertices

Ohba has conjectured \cite{ohb} that if the graph $G$ has $2χ(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More precisely we obtain that for any $0<ε<1$, there exist an $n_0=n_0(ε)$ such that the list chromatic number of $G$ equals its chromatic number, provided $$n_0 \leq |V(G) | \le (2-ε)χ(G).$$

List-edge-colouring planar graphs with precoloured edges

Let $G$ be a simple planar graph of maximum degree $Δ$, let $t$ be a positive integer, and let $L$ be an edge list assignment on $G$ with $|L(e)| \geq Δ+t$ for all $e \in E(G)$. We prove that if $H$ is a subgraph of $G$ that has been $L$-edge-coloured, then the edge-precolouring can be extended to an $L$-edge-colouring of $G$, provided that $H$ has maximum degree $d\leq t$ and either $d \leq t-4$ or $Δ$ is large enough ($Δ\geq 16+d$ suffices). If $d>t$, there are examples for any choice of $Δ$ where the extension is impossible.

无三角形平面图的列表着色

List colouring triangle free planar graphs

研究证明，无三角形平面图在独立集顶点给3种列表颜色、其余顶点给4种列表颜色时，总可进行列表着色。

Hyperbolicity Theorems for Correspondence Colouring

We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are criti…

The polynomial method for list-colouring extendability of outerplanar graphs

We restate theorems of Hutchinson on list-colouring extendability for outerplanar graphs in terms of non-vanishing monomials in a graph polynomial, which yields an Alon-Tarsi equivalent for her work. This allows to simplify her proofs as well as obtain more general results.

Colouring triangle-free graphs with local list sizes

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs. In both our results, we permit the amount of colour made available to vertices of lower degree to be accordingly lower. One result concerns list colouring and correspondence colouring, while the other concerns fractional colouring. Our proof of the second illustrates the use of the hard-core model to prove a Johansson-type result, which may be of independent interest.

关于多数边着色的列表扩展

On list extensions of the majority edge colourings

本文研究广义多数边着色的列表扩展，证明最小度满足相应二次或渐近界的图可从给定列表获得多数着色，并推广到颜色相关容差及非列表情形。

On the choosability with separation of planar graphs and its correspondence colouring analogue

A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for every $(\ell, k)$-list assignment $L$. We prove that if $G$ is a planar graph with $(4,2)$-list assignment $L$ and for every triangle $T \subseteq G$ we have that $|\bigcap_{v \in V(T)} L(v)| \neq 2$, then $G$ is $L$-colourable. In fact, we prove a slightly stronger result: if $G$ contains a clique $H$ such that $V(H) \cap V(T) \neq \emptyset$ for every triangl…

Perfect colourings of regular graphs

A vertex colouring of some graph is called perfect if each vertex of colour $i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes several restrictions on the colour incidence matrix $(a_{ij})$. We list several (old and new) necessary conditions for a matrix to be the colour incidence matrix of a perfect colouring. Moreover we show that a certain combination of these conditions is also sufficient. Using this we determine a list of all colour incidence matrices corresponding to perfect colourings of 3-regular, 4-regular and 5-regular graphs with two, three and four colours, re…

完全二部图中的装填着色与对应装填的逆问题

Packing colourings in complete bipartite graphs and the inverse problem for correspondence packing

本文研究非对称完全二部图的对应装填数与列表装填数，利用拉丁方刻画其取值，并证明除3外任意正整数均可作为对应装填数，同时反驳相关猜想。

Multiple list colouring of $3$-choice critical graphs

A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A characterization of $3$-choice critical graphs was given by Voigt in [On list Colourings and Choosability of Graphs, Habilitationsschrift, Tu Ilmenau(1998)]. Voigt conjectured that if $G$ is a bipartite $3$-choice critical graph, then $G$ is $(4m, 2m)$-choosable for every integer $m$. This conjecture was disproved by Meng, Puleo and Zhu in [On (4, 2)-Choosable Graphs, Journal of Graph Theory 85(2):412-428(2017)]. They showed that if $G=Θ_{r,s,t}$ where $r,s,t$ have the same parity …

Countable graphs are majority 3-choosable

The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but it is known that a 3-colouring with this property always exists. Anholcer, Bosek and Grytczuk recently gave a list-colouring version of this conjecture, and proved that such a colouring exists for lists of size 4. We improve their result to lists of size 3; the proof extends to directed acyclic graphs. We also discuss some generalisations.

图的列表区分指数

List distinguishing index of graphs

该研究探讨图的列表区分指数问题，证明除少数例外，当边的颜色列表大小至少为最大度Δ减1时，存在区分性边着色，该界对Δ≥3最优且适用于有限与无限图。

Equitable $d$-degenerate choosability of graphs

Let ${\mathcal D}_d$ be the class of $d$-degenerate graphs and let $L$ be a list assignment for a graph $G$. A colouring of $G$ such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a $d$-degenerate graph is called the $(L,{\mathcal D}_d)$-colouring of $G$. For a $k$-uniform list assignment $L$ and $d\in\mathbb{N}_0$, a graph $G$ is equitably $(L,{\mathcal D}_d)$-colorable if there is an $(L,{\mathcal D}_d)$-colouring of $G$ such that the size of any colour class does not exceed $\left\lceil|V(G)|/k\right\rceil$. An equitable $(L…

List colouring with a bounded palette

Král' and Sgall (2005) introduced a refinement of list colouring where every colour list must be subset to one predetermined palette of colours. We call this $(k,\ell)$-choosability when the palette is of size at most $\ell$ and the lists must be of size at least $k$. They showed that, for any integer $k\ge 2$, there is an integer $C=C(k,2k-1)$, satisfying $C = O(16^{k}\ln k)$ as $k\to \infty$, such that, if a graph is $(k,2k-1)$-choosable, then it is $C$-choosable, and asked if $C$ is required to be exponential in $k$. We demonstrate it must satisfy $C = Ω(4^k/\sqrt{k})$. For an integer $\ell…

Colouring of plane graphs with unique maximal colours on faces

The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1,...,4 in such a way that every face contains a unique vertex coloured with the maximal color appearing on that face. They proved that every plane graph has such a colouring with the numbers 1,...,6. We prove that every plane graph has such a colouring with the numbers 1,...,5 and we also prove the list variant of the statement f…

Colouring complete bipartite graphs from random lists

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side. For each vertex draw uniformly at random a list of size $k$ from a base set $S$ of size $s=s(n)$. In this paper we estimate the asymptotic probability of the existence of a proper colouring from the random lists for all fixed values of $k$ and growing $n$. We show that this property exhibits a sharp threshold for $k\geq 2$ and the location of the threshold is precisely $s(n)=2n$ for $k=2$, and approximately $s(n)=\frac{n}{2^{k-1}\ln 2}$ for $k\geq 3$.

List colourings of multipartite hypergraphs

Let $χ_l(G)$ denote the list chromatic number of the $r$-uniform hypergraph~$G$. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if $G$ is simple and $d$-regular, then $χ_l(G)\ge (1/(r-1)+o(1))\log_r d$. To see how close this inequality is to best possible, we examine $χ_l(G)$ when $G$ is a random $r$-partite hypergraph with $n$ vertices in each class. The value when $r=2$ was determined by Alon and Krivelevich, here we show that $χ_l(G)= (g(r,α)+o(1))\log_r d$ almost surely, where $d$ is the expected average degree of~$G$ and $α…

Improper choosability and Property B

A fundamental connection between list vertex colourings of graphs and Property B (also known as hypergraph 2-colourability) was already known to Erdős, Rubin and Taylor. In this article, we draw similar connections for improper list colourings. This extends results of Kostochka, Alon, and Král' and Sgall for, respectively, multipartite graphs, graphs of large minimum degree, and list assignments with bounded list union.

A note on two conjectures that strengthen the four colour theorem

There are two conjectures concerning planar graph colourings that are strengthenings of the four colour theorem. One concerns signed graph colouring and is proposed by Máčajová, Raspaud and Škoviera. It asserts that every signed planar graph is $4$-colourable. Another concerns list colouring and is proposed by Kündgen and Ramamurthi which asserts that if $L$ is a $2$-list assignment of a planar graph $G$, then there is an $L$-colouring of $G$ such that each colour class induces a bipartite graph. In this note we prove that the first conjecture implies the second one.

Colouring powers of cycles from random lists

Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the vertices of $C_n^k$ are those of the $n$-cycle, and two vertices are connected by an edge if their distance along the cycle is at most $k$). For each vertex draw uniformly at random a subset of size $c$ from a base set $S$ of size $s=s(n)$. In this paper we solve the problem of determining the asymptotic probability of the existence of a proper colouring from the lists for all fixed values of $c,k$, and growing $n$.

Asymptotically optimal bound on the adjacent vertex distinguishing edge choice number

An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from any set of lists associated to the edges of $G$ as long as the size of every list is at least $Δ+CΔ^{\frac{1}{2}}(\logΔ)^4$, where $Δ$ is the maximum degree of $G$ and $C$ is a constant. The proof is probabilistic. The same is true in the environment of total colourings.

A complexity trichotomy for approximately counting list H-colourings

We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem wh…

从列表中区分正则图

Distinguishing regular graphs from lists

该研究针对图的可区分边着色问题，证明了阶数≥7的至多可数连通正则图可由长度为2的任意列表集合实现可区分边着色，且该结论对阶为阿列夫分层不动点的连通正则图也成立。

The Thue choice number versus the Thue chromatic number of graphs

We say that a vertex colouring $φ$ of a graph $G$ is nonrepetitive if there is no positive integer $n$ and a path on $2n$ vertices $v_{1}\ldots v_{2n}$ in $G$ such that the associated sequence of colours $φ(v_{1})\ldotsφ(v_{2n})$ satisfy $φ(v_{i})=φ(v_{i+n})$ for all $i=1,2,\dots,n$. The minimum number of colours in a nonrepetitive vertex colouring of $G$ is the Thue chromatic number $π(G)$. For the case of vertex list colourings the Thue choice number $π_{l}(G)$ of $G$ denotes the smallest integer $k$ such that for every list assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ with minimum list len…

平面图的广义全染色

On Generalized Total Colourings of Planar Graphs

本文研究图的广义全染色，证实了Borowiecki等人提出的平面图广义全染色仅需4种颜色的猜想，针对两类无限平面图族完成了验证。

Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs

A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots \}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a subset of $\{1, 2, \dots\}$ of size $k$. A colouring $c$ of $G$ respects a $k$-regular list assignment $L$ of $G$ if $c(u)\in L(u)$ for every $u\in V$. A graph $G$ is $k$-choosable if for every $k$-regular list assignment $L$ of $G$, there exists a colouring of $G$ that respects $L$. We may also ask if for a given $k$-regular list assignment $L$ of a given g…

四维和五维中少色单纯形与超立方体的完美着色

Perfect colourings of simplices and hypercubes in dimension four and five with few colours

本文研究图的完美着色，完整确定四维与五维超立方体边图的两色、三色完美着色，并列出对应维数单纯形边图的所有完美着色以供比较。

On-line list colouring of random graphs

In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C in [2,4], depending on the range of d. Also, for d=O(1…

Packing 列表着色

Packing list-colourings

本文发起了对列表着色自然强化版本 list-packing 的研究，利用组合与概率方法为保证其存在的最小 k 设定了基于顶点数、退化度、最大度或列表色数的基本上界，并深入探讨了二部图情形及问题推广。

Narrowing the Complexity Gap for Colouring ($C_s$,$P_t$)-Free Graphs

For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a $k$-colouring of $G$ exists. The $k$-Precolouring Extension problem is to decide, for a given $G=(V,E)$, whether a colouring of a subset of $V$ can be extended to a $k$-colouring of $G$. A $k$-list assignment of a graph is an allocation of a list -a subset of $\{1,\ldots,k\}$- to each vertex, and the List $k$-Colouring problem is to decide, for a given $G$, whether $…

列表着色的重构

Reconfiguration of List Colourings

本文研究图的列表着色重构，证明在每点列表大小至少为度数加一且最大度不小于3时，任意两个可重着色着色可在平方步内转换，并揭示列表减小导致的尖锐相变。

The optimal edge-colouring threshold

Consider any dense r-regular quasirandom bipartite graph H with parts of size n and fix a set of r colours. Let L be a random list assignment where each colour is available for each edge of H with probability p. We show that the threshold probability for H to have a proper L-edge-colouring is p of order (log n)/n. This answers a question of Kang, Kelly, Kühn, Methuku and Osthus. We thus obtain the same threshold for Steiner Triple Systems and Latin squares; the latter answers a question of Johanssen from 2006.

List $k$-Colouring $P_t$-Free Graphs: a Mim-width Perspective

A colouring of a graph $G=(V,E)$ is a mapping $c\colon V\to \{1,2,\ldots\}$ such that $c(u)\neq c(v)$ for every two adjacent vertices $u$ and $v$ of $G$. The {\sc List $k$-Colouring} problem is to decide whether a graph $G=(V,E)$ with a list $L(u)\subseteq \{1,\ldots,k\}$ for each $u\in V$ has a colouring $c$ such that $c(u)\in L(u)$ for every $u\in V$. Let $P_t$ be the path on $t$ vertices and let $K_{1,s}^1$ be the graph obtained from the $(s+1)$-vertex star $K_{1,s}$ by subdividing each of its edges exactly once.Recently, Chudnovsky, Spirkl and Zhong (DM 2020) proved that List $3$-Colouring…

On the facial Thue choice index via entropy compression

A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge colouring of a path is nonrepetitive if the sequence of colours of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colourings for arbitrarily long paths using only three colours. A recent generalization of this concept implies that we may obtain such colourings even if we are forced to choose edge colours from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these…

List colouring of graphs and generalized Dyck paths

The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume $G$ is a graph and $f:V(G) \to N$ is a mapping. For a nonnegative integer $m$, let $f^{(m)}$ be the extension of $f$ to the graph $ G \diamondplus \overline{K_m}$ for which $f^{(m)}(v)=|V(G)|$ for each vertex $v$ of $\overline{K_m}$. Let $m_c(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $ G \diamondplus \overline{K_m}$ is not $f^{(m)}…

Colouring of generalized signed planar graphs

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We say $G$ is $S$-$k$-colourable if for any mapping $σ: E(G) \to S$ with $σ(x,y) = (σ(y,x))^{-1}$, there is a mapping $f: V(G) \to [k]=\{1,2, \ldots, k\}$ such that for each arc $e=(x,y)$, $σ_e(f(x)) \ne f(y)$. The concept of $S$-$k$-colouring is a common generalization of many colouring concepts, including $k$-colouring, signed $k$-colouring defined by Máčajová…

Every planar graph is $1$-defective $(9,2)$-paintable

Assume $L$ is a $k$-list assignment of a graph $G$. A $d$-defective $m$-fold $L$-colouring $ϕ$ of $G$ assigns to each vertex $v$ a set $ϕ(v)$ of $m$ colours, so that $ϕ(v) \subseteq L(v)$ for each vertex $v$, and for each colour $i$, the set $\{v: i \in ϕ(v)\}$ induces a subgraph of maximum degree at most $d$. In this paper, we consider on-line list $d$-defective $m$-fold colouring of graphs, where the list assignment $L$ is given on-line, and the colouring is constructed on-line. To be precise, the $d$-defective $(k,m)$-painting game on a graph $G$ is played by two players: Lister and Painter…

Maximum number of colourings. I. 4-chromatic graphs

It is proved that every connected graph $G$ on $n$ vertices with $χ(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed from $K_4$ by repeatedly adding leaves. This confirms (a strengthening of) the $4$-chromatic case of a long-standing conjecture of Tomescu [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets etiquetes, C. R. Acad. Sci. Paris 273 (1971), 1124-1126]. Proof methods may be of independent interest. In particular, one of our auxiliary results about …

基于随机列表的图着色

Colouring graphs from random lists

本文研究随机列表着色，证明最大度为 $\Delta$ 的图在 $m$ 满足相应条件时渐近几乎必然 $(k,m)$-可着色，并分析 $H$-free 图的阈值，证实并改进 Casselgren 的相关结论。

A Unified Approach to Distance-Two Colouring of Graphs on Surfaces

In this paper we introduce the notion of $Σ$-colouring of a graph $G$: For given subsets $Σ(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $Σ(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of these two colourings. …

Adaptable and conflict colouring multigraphs with no cycles of length three or four

The adaptable choosability of a multigraph $G$, denoted $\mathrm{ch}_a(G)$, is the smallest integer $k$ such that any edge labelling, $τ$, of $G$ and any assignment of lists of size $k$ to the vertices of $G$ permits a list colouring, $σ$, of $G$ such that there is no edge $e = uv$ where $τ(e) = σ(u) = σ(v)$. Here we show that for a multigraph $G$ with maximum degree $Δ$ and no cycles of length 3 or 4, $\mathrm{ch}_a(G) \leq (2\sqrt{2}+o(1))\sqrt{Δ/\lnΔ}$. Under natural restrictions we can show that the same bound holds for the conflict choosability of $G$, which is a closely related parameter…

着色问题中的同伦与超越扩张

Homotopies and transcendental extensions in colouring problems

本文通过代数无关坐标的几何实现与有向体积方法，证明了广义Y博弈不可绘制性等新结果，并重新证明了Sperner引理多重标记版本等已知结论。

Colouring Edges with many Colours in Cycles

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|; p + 1) colours. In the particular case where G has girth at least p + 1, Arb_p(G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring b…

Local girth choosability of planar graphs

In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a \emph{local girth list assignment}: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thoma…

最优地重构列表着色与对应着色

Optimally Reconfiguring List and Correspondence Colourings

本文研究列表着色与对应着色的重构直径，提出在列表大小至少为度数加2时的最优直径猜想，给出匹配构造，并证明更宽上界及高列表大小条件下的猜想，还验证了多类图。