Kempe equivalence of $4$-critical planar graphs

某些平面三角剖分的4-染色的Kempe等价

Kempe equivalence of 4-colourings of some plane triangulations

本文研究了具有特定度数条件的平面三角剖分图$G_n$的4-染色问题，枚举了其Kempe类和不同4-染色的数量，并证明了去除极点后的子图$H_n$的所有4-染色在有限次Kempe变换内等价。

Kempe等价与二次托里克环

Kempe equivalence and quadratic toric rings

本文通过Kempe等价与二次稳定集环的关系，揭示了图的稳定集环二次性的判定条件，并将其与完美可缩图理论联系起来，提出新的组合猜想。

On edge-colorings of bicubic planar graphs

In the first part, we introduce a notion a degree of edge-colorings of bicubic plane graphs and proves some local formula of the graded number of colorings. In the second part, we give a new proof of a result of Fisk saying that any two edge-3-colorings of a planar bicubic graph are Kempe equivalent. Additionaly we show that the degree of colorings behaves well with respect to Kempe equivalence.

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Let $G$ be a graph with a vertex colouring $α$. Let $a$ and $b$ be two colours. Then a connected component of the subgraph induced by those vertices coloured either $a$ or $b$ is known as a Kempe chain. A colouring of $G$ obtained from $α$ by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of $G$ are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. A conjecture of Mohar (2007) asserts that, for $k \geq 3$, all $k$-colourings of a $k$-regular graph that is not complete are Kempe equivalen…

平面图的Kempe等价列表边着色

Kempe Equivalent List Edge-Colorings of Planar Graphs

本文研究了具有较大最大度的平面图线图的 $L$-可交换性，证明了在 $Δ(G)\ge 9$ 且为 $(Δ(G)+1)$-分配，或 $Δ(G)\ge 15$ 且为 $Δ(G)$-分配时线图均 $L$-可交换。

Switching 3-edge-colorings of cubic graphs

The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders $4n+2$ and $4n+4$ with $2^n$ edge-Kempe equivalence classes are presented; it is conjectured that there are no cubic graphs with more edge-Kempe equivalence classes. New families of nonplanar bipartite cubic graphs with exactly one edge-Kempe equiva…

多面体连杆机构的普适性

Universality of Polyhedral Linkages

本文针对工程应用驱动的连杆机构研究，将平面连杆机构的Kempe普适性定理推广到三维及以上维度的刚性面多面体连杆机构，给出了嵌入构造方案。

Kempe 类与近似二部图

Kempe Classes and Almost Bipartite Graphs

本文研究近似二部图的 k-着色 Kempe 等价类，给出保证唯一等价类的新增边数阈值，并构造反例证明其渐近紧、反驳相关猜想。

The Birkhoff Diamond as Double Agent

Despite the existence of a proof of the 4-color theorem, it would seem that there is still more to learn about why any planar graph is 4-colorable. To that end, we take another look at the Birkhoff diamond and discover something new and intriguing: after an extensive search for (rare) Kempe-locked triangulations, we find a Birkhoff diamond subgraph in each one. We offer a heuristic argument as to why that result is not only reasonable but also to be expected and posit that the presence of a Birkhoff diamond is necessary to Kempe-locking. If that conjecture is true, it means that the Birkhoff d…

Hamiltonian cycles and 1-factors in 5-regular graphs

It is proven that for any integer $g \ge 0$ and $k \in \{ 0, \ldots, 10 \}$, there exist infinitely many 5-regular graphs of genus $g$ containing a 1-factorisation with exactly $k$ pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle. For $g = 0$, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's "marriage" operation, we discuss two gluing techniques aimed at producing graphs of high cyclic edge-connectivity. We prove that there exist infinitely many planar 5-connected 5-regular graphs in which every 1-factorisation has zero perfect pairs. On the other…

Strengthening a theorem of Meyniel

For an integer $k \geq 1$ and a graph $G$, let $\mathcal{K}_k(G)$ be the graph that has vertex set all proper $k$-colorings of $G$, and an edge between two vertices $α$ and~$β$ whenever the coloring~$β$ can be obtained from $α$ by a single Kempe change. A theorem of Meyniel from 1978 states that $\mathcal{K}_5(G)$ is connected with diameter $O(5^{|V(G)|})$ for every planar graph $G$. We significantly strengthen this result, by showing that there is a positive constant $c$ such that $\mathcal{K}_5(G)$ has diameter $O(|V(G)|^c)$ for every planar graph $G$.

一种无需计算机证明四色定理的革新方法，第一部分：极大平面图上的 RGB-铺砌

A renewal approach to prove the Four Color Theorem unplugged, Part I: RGB-tilings on maximal planar graphs

本文提出不用计算机检验证明四色定理的三部分新方案，首篇在极大平面图上建立R/G/B与RGB边铺砌及其与顶点4-着色的联系。

关于特征三平面函数的一个分类

On a classification of planar functions in characteristic three

本文系统研究了特征三有限域上平面函数的CCZ-等价与同位等价分类，完全解决了$n\le 11$时已知平面函数的同位等价是否产生新CCZ-类的问题，并在$\mathbb{F}_{3^6}$和$\mathbb{F}_{3^9}$上发现了九个新的零散平面函数。

On graph equivalences preserved under extensions

Let R be an equivalence relation on graphs. By the strengthening of R we mean the relation R' such that graphs G and H are in the relation R' if for every graph F, the union of the graphs G and F is in the relation R with the union of the graphs H and F. We study strengthenings of equivalence relations on graphs. The most important case that we consider concerns equivalence relations defined by graph properties. We obtain results on the strengthening of equivalence relations determined by the properties such as being a k-connected graph, k-colorable, hamiltonian and planar.

关于弯曲函数和平面函数的一些观察

Some observations on bent and planar functions

本文研究了弯曲函数和平面函数的性质，证明了其图的Salem集性质，并展示了平面函数间距离的下限为二。

三树定理

The Three Tree Theorem

证明不同于棱柱的2-球面图可4着色使Kempe链为森林，得出离散2-球面树荫度为3，且三棵树可覆盖所有三角形，推论平面图树荫度≤3。

Do triangle-free planar graphs have exponentially many 3-colorings?

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $α$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset $A'$ of $A$ of size at least $α|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we class…

Testing Mutual Duality of Planar Graphs

We introduce and study the problem \mpd, which asks for two planar graphs $G_1$ and $G_2$ whether $G_1$ can be embedded such that its dual is isomorphic to $G_2$. Our algorithmic main result is an NP-completeness proof for the general case and a linear-time algorithm for biconnected graphs. To shed light onto the combinatorial structure of the duals of a planar graph, we consider the \emph{common dual relation} $\sim$, where $G_1 \sim G_2$ if and only if they have a common dual. While $\sim$ is generally not transitive, we show that the restriction to biconnected graphs is an equivalence relat…

Feynman identity for planar graphs

The Feynman identity (FI) of a planar graph relates the Euler polynomial of the graph to an infinite product over the equivalence classes of closed nonperiodic signed cycles in the graph. The main objectives of this paper are to compute the number of equivalence classes of nonperiodic cycles of given length and sign in a planar graph and to interpret the data encoded by the FI in the context of free Lie superalgebras. This solves in the case of planar graphs a problem first raised by S. Sherman and sets the FI as the denominator identity of a free Lie superalgebra generated from a graph. Other…

$K_5$-无 minors 图的度向割半群饱和性与 Seymour 平面边染色猜想

Degreewise Cut-Semigroup Saturation for $K_5$-Minor-Free Graphs and Seymour's Planar Edge-Colouring Conjecture

该论文证明了 $K_5$-无 minors 图的齐次割半群在高度 $k$ 的饱和性与平面 $k$-图的 $k$-边可染色性等价，从而将割多面体正规性猜想与 Seymour 平面边染色猜想联系起来。

Discrete Trace Theorems and Energy Minimizing Spring Embeddings of Planar Graphs

Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. It also follows fairly quickly that this embedding minimizes the sum of squared edge lengths, conditional on the embedding of the outer face. However, it is not at all clear how to embed this outer face. We consider the minimization problem of embedding this outer face, up to som…

关于$r$-图与等价性的一些猜想

Some conjectures on $r$-graphs and equivalences

本文研究Seymour关于平面$r$-图边可着色性和完美匹配覆盖的猜想，并证明在$t,r\geq1$时，平面$r$-图、无$K_5$ minors的$r$-图、无$K_{3,3}$ minors的$r$-图以及交叉数至多为1的$r$-图具有$(t,r)$-完美匹配这四种陈述等价。

A note on geometric duality in matroid theory and knot theory

We observe that for planar graphs, the geometric duality relation generates both 2-isomorphism and abstract duality. This observation has the surprising consequence that for links, the equivalence relation defined by isomorphisms of checkerboard graphs is the same as the equivalence relation defined by 2-isomorphisms of checkerboard graphs.

平面复形上的轨道与sofic性

Tracks on planar complexes and soficity

本文证明由具有平面连通分量的局部有限Borel图生成的保测等价关系是sofic的，从而去除了先前工作中关于单模随机平面图sofic性的额外假设。

平面站点渗流、端结构以及Benjamini-Schramm猜想

Planar Site Percolation, End Structure, and the Benjamini-Schramm Conjecture

本文研究平面图中站点渗流的临界阈值与唯一性，证明在可数端等价类下共存区间内不存在唯一性，同时构造了反例说明Benjamini-Schramm猜想不成立。

拟等距、收缩与交图

Quasi-isometries, contractions, and intersection graphs

本文证明图为拟平面图当且仅经两类有界迭代操作得到，推广了字符串图为拟平面图的结论，还推导了拟平面图的收缩子式性质及树分解相关结果，为拟等距等提供了工具。