围长为6的三次图的圆边染色

Three-coloring triangle-free graphs on surfaces III. Graphs of girth five

We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.

着色、稀疏和周长

Coloring, sparseness, and girth

一个$r$扩充树是一个有根树加上从每个叶子到祖先添加的$r$边。对于$d,g,r\in\mathbb{N}$，我们构造了一棵围长至少为$g$的二分$r$ -增广完全$d$ -二叉树.这种树的高度必须以极快的速度增长。利用所得到的图，我们构造了稀疏的非$k$ -可选的二部图，证明了最大平均度至多为$2(k-1)$是二部图中$k$ -可选的一个尖锐的充分条件，即使在要求大围长的情况下也是如此.我们还给出了具有任意围长$g$的非$k$ -可着色图和超图的一个新的简单构造.

3列出曲面上围长至少为5的着色图

3 List Coloring Graphs of Girth at least Five on Surfaces

Grotzsch证明了每个无三角形的可平面图都是3-可着色的。Rassen证明了围长至少为5的可平面图是3-可选图。对于其它曲面，Kassen证明了在任意固定曲面中，围长至少为5的4-临界图只有200个。这意味着一个线性时间的算法，以确定3-着色的围长至少为5的任何固定的表面上的图。Dvorak，Kral和Thomas通过证明围长至少为5的4-临界图的顶点数在其亏格中是线性的，加强了Dvorassen的结果。他们用这个结果证明了哈维尔猜想，即一个三角形两两相距足够远的可平面图是3-可着色的。关于列表染色，Dvorak证明了一个可平面图，如果它的圈数不超过4，则它是3-可选的。本文推广了这些结果。首先，我们证明了围长至少为5的3-列表着色图的线性等周界。许多新的结果，然后遵循理论的双曲族的图开发的波斯特尔和托马斯。特别地，在任意固定曲面上，围长至少为5的4-链表-临界图只有1000个，而且4-链表-临界图的顶点数在其亏格中是线性的.这提供了上述结果的独立证明，同时将Dvorak的结果推广到具有大边宽度的曲面上的图，并产生类似的结果，表明周长至少为5且交叉点成对相距很远的图是3-choosable的。最后，我们推广到曲面上，证明了围长至少为5的可平面图有指数级多个不同的3-列表染色。具体地说，我们证明了每个围长至少为5且具有3-列表着色的图都具有$2^{Ω(n)-O(g)}$不同的3-…

具有特定围长的可平面图的对策色数的界

Bounds for the game coloring number of planar graphs with a specific girth

设${\rm col_g}(G)$为给定图$G.$的对策色数定义图族$\mathcal{H}$为${\rm col_g}(\mathcal{H}) := \max\{{\rm col_g}(G):G \in \mathcal{H}\}.$设$\mathcal{P}_k$为围长至少为$k.$的可平面图族我们证明了${\rm col_g}(\mathcal{P}_7) \leq 5.$这一结果推广了Wang和Zhang {WZ 11}（我们还证明了这些边界是尖锐的，通过构建一个图$G$，其中$G \in {\rm col_g}(\mathcal{P}_k) \geq 5$对于每个$k \leq 8$，使得${\rm col_g}(G)=5.$作为结果，${\rm col_g}(\mathcal{P}_k) = 5$对于$k =7,8.$

关于围长为5的平面图2-距离着色的若干结果

Some results on 2-distance coloring of planar graphs with girth five

针对围长为5的平面图的2-距离着色问题，本文证明其存在2-距离$Δ+7$着色（改进已有结果），且当最大度$Δ\geq 10$时存在2-距离$Δ+6$着色，为该类图的着色研究提供了更优结论。

关于循环多面体的无冲突着色与图的围长猜想

On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs

本文研究循环多面体对应超图的无冲突色数，给出若干偶数维的精确渐近界及一般偶数维上下界，并揭示其与 Erdős 围长猜想的联系。

可平面图的强边染色

Strong edge coloring of planar graphs

图的强边染色是指距离至多为2的边具有不同颜色的真边染色。已知任意最大度为D的可平面图都有至多4D + 4色的强边染色。我们表明，3D + 6颜色足够的，如果图的围长为6，和3D颜色足够的，如果围长至少是7。此外，我们证明了围长至少为6的立方可平面图可以被最多9种颜色的强边染色。

Fractional coloring of planar graphs of girth five

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex x of G, the graph G has a set coloring phi by subsets of {1,...,6} such that |phi(v)|>=2 for each vertex v of G and |phi(x)|=3. As a corollary, every triangle-free planar graph on n vertices is (6n:2n+1)-colorable. We further use this result to prove that for every Delta, there exists a constant M_Delta such that every planar graph G of girth at least five a…

高围长图的${\rm b}^*$-着色与$z$-着色

On ${\rm b}^{\ast}$-Coloring and $z$-Coloring of graphs with high girth

该研究针对高围长图，运用列表着色技术推导了${\rm b}^*$-色数、$z$-色数的相关结论，丰富了图着色理论的成果。

大围长三次图的控制数

Domination number of cubic graphs with large girth

证明了围长不小于g的n-顶点三次图的控制数不超过0.299871n + O（n/g）<3n/10 + O（n/g）.

局部圈长对k饱和子三次图S-打包着色的影响

Impact of local girth on the S-packing coloring of k-saturated subcubic graphs

研究局部圈长对k饱和子三次图S-打包着色能力的影响，通过参数分析和构造验证，揭示饱和度与局部圈长共同决定的可接受S-打包序列。

围长等于度数加1的正则图的全染色

Total coloring of regular graphs of girth = degree + 1

本文研究围长等于度数加1的正则图的高效全染色，构造了围长为4的三次图的高效全染色并提出四种基本运算猜想，同时给出若干非高效全染色实例及对3-路和3-星划分的应用。

具有唯一最长环的三次图的围长与连通性研究

On the girth and connectivity of cubic graphs with a unique longest cycle

该研究证明存在无穷多族围长为5的2-连通非哈密顿三次图，且这类图都包含唯一最长环，为三次图的结构性质研究提供了新结论。

Complexity of acyclic colorings of graphs and digraphs with degree and girth constraints

We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from…

围长至少为21的平面次立方图的2-距离4-染色

2-distance 4-coloring of planar subcubic graphs with girth at least 21

本文证明了围长至少为21的平面次立方图具有2-距离4-染色，并构造出围长为11的平面次立方图反例，界定了该染色性质成立的围长下界。

On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons

An edge coloring of a simple graph $G$ is said to be \textit{proper rainbow-cycle-forbidding} (PRCF, for short) if no two incident edges receive the same color and for any cycle in $G$, at least two edges of that cycle receive the same color. A graph $G$ is defined to be \textit{PRCF-good} if it admits a PRCF edge coloring, and $G$ is deemed \textit{PRCF-bad} otherwise. In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4. They then ask whether such graphs exist having girth greater than 4. In our work, we give…

The girths of the cubic Pancake graphs

The Pancake graphs $P_n, n\geqslant 2$, are Cayley graphs over the symmetric group $\mathrm{Sym}_n$ generated by prefix-reversals. There are six generating sets of prefix-reversals of cardinality three which give connected Cayley graphs over the symmetric group known as cubic Pancake graphs. In this paper we study the girth of the cubic Pancake graphs. It is proved that considered cubic Pancake graphs have the girths at most twelve.

高围长三次图与Clebsch图同态

High-girth cubic graphs are homomorphic to the Clebsch graph

我们给出了一个（计算机辅助）证明，每个图的最大度为3，围长至少为17的边缘可能是5色的（可能不正确），使每个颜色类的补充是二分的。等价地，每一个这样的图承认一个同态的Clebsch图。Hopkins、Staton、Bondy和Locke证明了每一个围长至少为4的（次）三次图都有一个至少包含4/5条边的边割。这样的边割的存在性直接来自于上面描述的5-边着色的存在性，所以我们的定理可以被看作是他们结果的着色扩展（在更强的围长假设下）。每个与长度为5的圈同态的图都有上述的5-边染色;因此我们的定理也可以被看作是Nesetril的五角大楼问题的一个弱版本（该问题询问是否每个足够高围长的立方图都与C_5同态）。

围长为6的次立方可平面图的内射可选性

Injective choosability of subcubic planar graphs with girth 6

图$G$的内射着色是指对图$G$的顶点进行颜色分配，使得具有公共邻居的任意两个顶点具有不同的颜色。一个图$G$是$k$ -可内射选择的，如果对任何列表分配$L$，其中$|L(v)| \geq k$对所有$v \in V(G)$，$G$有一个单射$L$ -着色.内射着色在纠错码理论中有应用，并且与其他可着色性概念密切相关。本文证明了围长至少为6的次立方可平面图是5-可内射选择图。这加强了Lužar、Škrekovski和Tancer的一个结果，即围长至少为7的次立方可平面图是5-可注入着色的。我们的结果也改进了其他几个结果在特定情况下。

给定围长的立方点传递图

On cubic vertex-transitive graphs of given girth

针对立方点传递图的可区分代价研究空白，本文证明有两个边轨道且围长为5的该类图可区分代价为2，且不存在围长为6的无限3-弧传递立方图。

周长至少为$5$的图的$(1, k)$ -染色

$(1, k)$-coloring of graphs with girth at least $5$ on a surface

一个图是$(d_1, ..., d_r)$ -可着色的，如果它的顶点集可以划分成$r$集$V_1, ..., V_r$，使得对于每个$i\in \{1, ..., r\}$，由$V_i$诱导的图的最大度至多为$d_i$.对于给定的$(g, d_1)$对，确定最小的$d_2=d_2(g; d_1)$使得围长至少为$g$的平面图是$(d_1, d_2)$ -可着色的问题引起了人们的极大兴趣.除$(g, d_1)=(5, 1)$外，所有情况下都知道$d_2(g; d_1)$的有限性。Montassier和Ochem明确地询问$d_2(5; 1)$是否是有限的。我们回答这个问题的肯定与$d_2(5; 1)\leq 10$，即，我们证明了所有的平面图周长至少$5$是$(1, 10)$ -着色。此外，我们的证明扩展到声明，任何表面的欧拉属$γ$ $S$，存在一个$K=K(γ)$，其中周长至少$5$是嵌入$S$的图形是$(1, K)$ -着色。另一方面，不存在有限$k$，其中围长至少为$5$的平面图（因此可嵌入任何曲面）是$(0, k)$ -可着色的。

围长至少为7的稀疏图的正常无冲突可选择性

Proper conflict-free choosability of sparse graphs with girth at least seven

本文证明围长至少7且最大平均度小于8/3的图可正常无冲突（度+2）选择着色，并改进平面图围长界至8。

平面图的不相交列表着色

Disjoint list-colorings for planar graphs

本研究证明平面图围长至少8时存在两个不相交的3-列表着色，围长至少6时存在分数打包，并优化了列表打包数与ε-灵活性界。

围长为5的亚Hamilton平面三次图

Hypohamiltonian planar cubic graphs with girth five

如果一个图不是哈密尔顿图，但如果去掉任何一个顶点，它就变成哈密尔顿图，则称它是次哈密尔顿图。从1981年Jassen构造平面三次图开始，人们发现了许多次哈密尔顿平面三次图。然而，到目前为止发现的所有例子都有4个周期。在这篇笔记中，我们提出了第一个例子，亚哈密尔顿平面三次图的循环连通度为5，从而围长为5。通过计算机搜索，我们证明了这类图的最小成员是三个有76个顶点的图。

大围长三次图和随机三次图的最大边割

Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs

我们证明了对于任意围长足够大的三次图G，G的边割存在一个概率分布，使得G的每条边以至少0.88672的概率处于随机选择的割中.这意味着G包含大小至少为1.33008 n的边割，其中n是G的顶点数，并且具有至多为1.127752的分数割覆盖数。最大边割的大小的下界也适用于随机三次图。具体地说，随机n-顶点立方图a.a.s.包含大小为1.33008n的边缘切割。

A Note on Coloring Digraphs of Large Girth

The digirth of a digraph is the length of a shortest directed cycle. The dichromatic number $\vecχ(D)$ of a digraph $D$ is the smallest size of a partition of the vertex-set into subsets inducing acyclic subgraphs. A conjecture by Harutyunyan and Mohar states that $\vecχ(D) \le \left\lceil\fracΔ{4}\right\rceil+1$ for every digraph $D$ of digirth at least $3$ and maximum degree $Δ$. The best known partial result by Golowich shows that $\vecχ(D) \le \frac{2}{5}Δ+O(1)$. In this short note we prove for every $g \ge 2$ that if $D$ is a digraph of digirth at least $2g-1$ and maximum degree $Δ$, then…

近着色：不可着色图与NP-完全性

Near-colorings: non-colorable graphs and NP-completeness

图G是（d_1，.，d_l）-可着色，如果G的顶点集可被划分为子集V_1，.，V_1，使得由V_i的顶点导出的图G[V_i]对所有1 <= i <= 1具有最大度至多为d_i。在本文中，我们专注于复杂性方面的这种着色时，l= 2，3。更精确地说，我们证明了：对于任意固定整数k，j，g，（k，j）不同形式（0，0）且g ≥ 3，或者每个围长至少为g的可平面图是（k，j）-可染的，或者判定围长至少为g的可平面图是否是（k，j）-可染的是NP-完全的.对于任意固定整数k，判定一个可平面图是（0，0，0）-可着色图还是非（k，k，1）-可着色图是（0，0，0）-可着色图是NP-完全的。此外，我们还证明了围长为5的非（3，1）-可着色可平面图和围长为7的非（2，0）-可着色可平面图.

$2$-distance list $(Δ+2)$-coloring of planar graphs with girth at least 10

Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say that a graph is list $k$-colorable when every vertex $v$ has a list of colors of size at least $k$. A $2$-distance coloring is a coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance list ($Δ+2$)-coloring for planar graphs with girth at least $10$ and maximum degree $Δ\geq 4$.

立方图的高效全着色及其相关主题

Face $S_3$-colorings and Petrie monochromatic circuits for efficient total arc colorings of cubic maps of girth 4

研究立方图的高效全着色方法，探讨girth为4的图的全着色特性，并提出相关猜想。

b-continuity and Partial Grundy Coloring of graphs with large girth

A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of $G$ is the set $S_{b}(G)$ of integers $k$ such that $G$ has a b-coloring with $k$ colors and $b(G)=\max S_{b}(G)$ is the b-chromatic number of $G$. A graph is b-continous if $S_{b}(G)=[χ(G),b(G)]\cap \mathbb{Z}$. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. A partial Grundy coloring is a proper coloring $f:V(G)\rightarrow \{1,\ldots,k\}$ such that e…

Counting independent sets in cubic graphs of given girth

We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minim…

Odd coloring of sparse graphs and planar graphs

An {\it odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petru\v sevski and \v Skrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph $G$ with $mad(G)\leq \frac{4c-4}{c+1}$ has an odd $c$-coloring for $c\geq 4$, and proved the conjecture for $c\in\{5, 6\}$. In particular, planar graphs with girth at least $7$ and $6$ have an odd $5$-…

具有不可避免的单色边或彩虹边的高围长超图

High girth hypergraphs with unavoidable monochromatic or rainbow edges

Erdens和Hajnal的一个经典结果是：对任意整数$k, r, g \geq 2$，存在围长至少为$g$且色数至少为$k$的$r$ -一致超图.这意味着存在稀疏超图，使得在其顶点的任何着色中，至多有$k-1$颜色，存在单色超边。证明了对任意整数$r, g\geq 2$，存在周长至少为$g$的$r$ -一致超图，使得其顶点的任意着色都有单色边或彩虹边.我们给出了一个概率和确定性证明这一结果。

An approach to the girth problem in cubic graphs

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph theorem, $E$ is the disjoint union of a $2$-factor and a perfect matching $M$. We refer to the edges of $M$ as chords and classify the cycles in $G$ by their number of chords. We define $γ_k(n)$ to be the largest integer $g$ such that every cubic $n$-vertex graph with a given perfect matching $M$ has a cycle of length at most $g$ with at most $k$ chords. Here we de…

围长为六的三次点传递图

Cubic vertex-transitive graphs of girth six

本文完整分类了围长为6的有限简单三次点传递图，证明除Desargues图外，它们均可由环面六边形平铺、闭双曲曲面弧传递三角剖分的截断或6-正则图的弧传递二面体方案截断得到。

关于girth至少为五的图实现Steffen边着色界的图

On graphs with girth at least five achieving Steffen's edge coloring bound

研究通过环图实现Steffen边着色界，证明特定条件下临界图必须为奇数girth的环图。

具有奇围长的强公共图是圈

Strongly common graphs with odd girth are cycles

本文回答了 Chen 和 Ma 的问题，证明了具有奇围长但非圈的图不是强公共图。

On Star 5-Colorings of Sparse Graphs

A \textit{star $k$-coloring} of a graph $G$ is a proper (vertex) $k$-coloring of $G$ such that the vertices on a path of length three receive at least three colors. Given a graph $G$, its \textit{star chromatic number}, denoted $χ_s(G)$, is the minimum integer $k$ for which $G$ admits a star $k$-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the \textit{maximum average degree}, denoted $mad(G)$, of a graph $G$ is $\max\left\{ \frac{2|E(H)|}{|V(H)|}:{H \subset G}\right\}$. It is known that for a grap…

大围长平面图的染色对策--稀疏仙人掌的一个结果

The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses

我们用$χ$ g（G）表示图G的游戏色数，它是Alice在G上赢得着色游戏所需的最小颜色数.我们从Montassier et al. [M. Montassier，P. Ossona de Mendez，A. Raspaud和X. Zhu，Decomposing a graph into forests，J. Graph Theory Ser. B，102（1）：38-52，2012]，并且独立地，从Wang和Zhang，[Y. Wang和Q.张某Decomposing a planar graph with girth at least 8 into a forest and a matching，Discrete Maths，311：844-849，2011] that planar graphs with girth at least 8 have game chromatic number at most 5.人们可能会问，对于足够大的围长，这个5的界是否可以改进。在本文中，我们证明它不能。不仅如此，我们还证明了存在仙人掌CT（即每条边最多只属于一个圈的图）的$χ$ g（CT）= 5，尽管其圈长任意大，甚至圈间距离任意大.

缺陷边着色的奇围长界

Odd-Girth Bounds for Defective Edge Coloring

该研究针对无环非二部多重图的缺陷边着色问题，证明了结合奇围长参数的上界公式，验证了几类特殊情形下的已知结论，构造了达到等号的几乎全环多重图，并推导了缺陷Goldberg--Seymour猜想的成立范围。

围长为6的可平面图分裂为两个短路线性森林

Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

最近，Borodin，Kostochka和Yancey（On $1$ -inappropriate $2$ -coloring of sparse graphs. Discrete Mathematics，313（22），2013）示出了围长至少为$7$的每个平面图的顶点可以是$2$着色的，使得每个颜色类导出匹配的子图。我们证明了任何周长至少为$6$的平面图都允许$2$颜色的顶点着色，使得每个单色分量是长度至多为$14$的路。此外，我们显示了这个结果的列表版本。另一方面，对于每个正整数$t\geq 3$，我们构造了一个周长$4$的平面图，使得在$2$颜色的顶点的任何着色中，存在长度至少为$t$的单色路径.围长$5$的每个平面图是否允许没有长单色路的$2$ -染色仍然是开放的。

Circular Coloring and Fractional Coloring in Planar Graphs

We study the following Steinberg-type problem on circular coloring: for an odd integer $k\ge 3$, what is the smallest number $f(k)$ such that every planar graph of girth $k$ without cycles of length from $k+1$ to $f(k)$ admits a homomorphism to the odd cycle $C_k$ (or equivalently, is circular $(k,\frac{k-1}{2})$-colorable). Known results and counterexamples on Steinberg's Conjecture indicate that $f(3)\in\{6,7\}$. In this paper, we show that $f(k)$ exists if and only if $k$ is an odd prime. Moreover, we prove that for any prime $p\ge 5$, $$p^2-\frac{5}{2}p+\frac{3}{2}\le f(p)\le 2p^2+2p-5.$$ …

大围长图的b-连续性

The b-continuity of graphs with large girth

一个图的顶点的b-染色是一个正常染色，其中每个颜色类包含一个顶点，该顶点与其他颜色类相邻。$G$的b-色数是使$G$具有带$b(G)$颜色的b-染色的最大整数$b(G)$。一个图$G$是b-连续的，如果对区间$[χ(G),b(G)]$中的每一个整数$k$，$G$都有一个用$k$颜色的b-染色.众所周知，并非所有的图都是b-连续的。在这篇文章中，我们证明了，如果$G$的围长至少为10，那么$G$是b-连续的。

围长为五的Ricci平坦立方图

Ricci-flat cubic graphs with girth five

本文分类了所有围长至少为5的连通简单3-正则Ricci平坦图，证明仅有Petersen图、Triplex和十二面体图，并更正了Lin-Lu-Yau在2014年遗漏Triplex的分类。

围长为七的三次点传递图

Cubic vertex-transitive graphs of girth seven

本文通过图的签名对围长为7的三次点传递图进行完整分类，并证明若每条边包含在相同数量的7-圈中，则该图也是弧传递的。

曲面上无三角形图的指数多4-列表着色

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

证明了平面图$G$在$n$顶点上至少有$2^{n/9}$不同的$L$ -着色，如果$L$是$G$的5-列表-赋值，至少有$2^{n/10000}$不同的$L$ -着色，如果$L$是$G$的3-列表-赋值，并且$G$的围长至少为5。Postle和Thomas证明了：如果$G$是一个图，它的$n$顶点嵌入在一个属$g$的曲面$Σ$上，则存在常数$ε,c_g > 0$使得如果$G$具有$L$着色，然后$G$有至少$c_g2^{εn}$不同的$L$ -着色如果$L$是$G$或如果$L$是$G$和$G$的3列表分配腰围至少有五更一般地，他们证明了存在常数$ε,α>0$，使得如果$G$是一个图，其$n$顶点嵌入在固定亏格$g$的表面$Σ$中，则$H$是$G$的真子图，并且$ϕ$是$H$的一个$L$ -着色，其扩展到$G$的一个$L$ -着色，则如果$L$是5-列表分配或者如果$L$是3-列表分配并且$G$具有至少5的围长，则$ϕ$扩展到$G$的至少$2^{ε(n - α(g + |V(H)|))}$不同的$L$着色。我们证明了同样的结果，如果$G$是三角自由和$L$是一个4-列表分配的$G$，其中$ε=\frac{1}{8}$，和$α= 130$。

稀疏亚立方图的(1,2⁴)和(1,2⁵)-填充边着色

On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs

本文针对亚立方平面图，确定了使这类图存在(1,2⁵)-、(1,2⁴)-填充边着色的最小girth阈值的范围，证明两阈值均为有限值。

Girth, oddness, and colouring defect of snarks

The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect $0$ if and only if it is $3$-edge-colourable, this invariant can measure how much a cubic graph differs from a $3$-edge-colourable graph. Our aim is to examine the relationship of colouring defect to oddness, an extensively studied measure of uncolourability of cubic graphs, defined as the smallest number of odd circuits in a $2$-factor. We show that there exist cyclically $5$-edge-connected snarks (cu…

Girth-regular graphs

We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the edges with end-vertex $u$ lie on. Girth-regularity generalises two very different aspects of symmetry in graph theory: that of vertex transitivity and that of distance-regularity. For general girth-regular graphs, we give some results on the extremal cases of signatures. We then focus on the cubic case and provide a characterisation of cubic girth-regular g…

大度数稀疏图平方的列表染色

List coloring the square of sparse graphs with large degree

我们考虑有界最大平均度图的平方着色问题，即在保证相邻或有共同邻居的两个顶点接收不同颜色的情况下对顶点着色的问题。Borodin等人在2004年和2008年证明了围长至少为7且最大度足够大的平面图$Δ$的平方是列表$(Δ+1)$ -可染的，而围长为6且最大度任意大的平面图的平方不是列表$(Δ+1)$ -可染的。根据欧拉公式，围长至少为$6$的平面图的最大平均度小于$3$，围长至少为$7$的平面图的最大平均度小于$14/5<3$ .我们加强了他们的结果，证明了存在一个函数$f$使得任何具有最大平均度$m<3$和最大度$Δ\geq f(m)$的图的平方都是列表$(Δ+1)$ -可染的. $3$的这一界是最优的，在这个意义上，上述平面图的周长$6$有最大平均度小于$3$和任意大的最大度，而他们的平方不能是$(Δ+1)$ -着色。这同样适用于列表单射$Δ$ -着色。

具有大围长的三次图和粗图的生成

Generation of cubic graphs and snarks with large girth

我们描述了两个新的算法生成的所有非同构的三次图的围长至少$k\ge 5$是非常有效的$5\le k \le 7$，并显示这些算法可以有效地限制生成的snark围长至少$k$。我们实现这些算法是超过30，分别比以前最快的生成器的周长至少为6和7，分别40倍。使用这些生成器，我们也产生了所有的非同构的咆哮周长至少6至38顶点，并表明，没有咆哮周长至少7至42顶点。我们提出并分析了周长为6的新的咆哮列表。

稀疏图的正常着色与平方着色之间的着色

Between proper and square colorings of sparse graphs

该研究探讨介于正常着色与平方着色之间的packing着色，构造了不满足对应着色的稀疏/平面图，确定了平面图的最小围长条件，并拓展了次立方图的列表着色结果、强化了非列表可着色的平面图示例。

围长为6的三次图有Petersen子式

Girth six cubic graphs have Petersen minors

本文证明了任何不含长度小于6的圈的3-正则图都有一个子图同构于Petersen图的一个剖分。

关于图的幂着色

On Colorings of Graph Powers

本文给出了关于图的幂着色的一些结果。引入了螺旋图的概念。我们证明了这类图对于其$(2t+1)$ st幂由Kneser图有界的高奇围长图是hom-universal的。此外，我们考虑了奇圈同态的存在性问题。证明了当且仅当$S_2(G)$的$(2k+1)$ 1次幂的色数小于或等于3时，这种与$(2k+1)$ -圈的同态存在，其中$S_2(G)$是$G$的2-细分.我们还考虑了奈舍特基尔的五角大楼问题。这个问题是关于是否存在不同态于大小为5的圈的高围长三次图。介绍了几个与Nešetzanil问题密切相关的问题，并给出了它们之间的关系。