George Grätzer
A planar semimodular lattice $L$ is \emph{slim} if $\mathbf{M}_3$ is not a sublattice of $L$. In a recent paper, G. Czédli introduced a very powerful diagram type for slim, planar, semimodular lattices. This short note proves the existence of such diagrams.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The swing lemma, due to G. Grätzer for slim semimodular lattices and extended by G. Czédli and G. Grätzer for all planar semimodular lattices, describes the congruence generated by a prime interval in an efficient way. Here we present a new proof for this lemma, which is shorter than the earlier two. Also, motivated by the swing lemma and mechanical pinball games with flippers, we construct an online game called Swing lattice game. A computer program realizing this game is available from the authors' websites.
Note on antichain cutsets in discrete semimodular lattices
Rival和Zaguia将有限布尔格的水平集刻画为反链割集，这在所有离散半模格中都成立。
A basic property in a modular lattice is that any two flags generate a distributive sublattice. It is shown (Abels 1991, Herscovic 1998) that two flags in a semimodular lattice no longer generate such a good sublattice, whereas shortest galleries connecting them form a relatively good join-sublattice. In this note, we sharpen this investigation to establish an analogue of the two-flag generation theorem for a semimodular lattice. We consider the notion of a modular convex subset, which is a subset closed under the join and meet only for modular pairs, and show that the modular convex hull of t…
A Note on Planar and Dismantlable Lattices
它表明，任何有限的，秩连接，可压缩格是字典序壳（因此科恩-麦考利）。一个等级，区间连接的格被证明是等级连接的，但等级连接的格不需要是区间连接的。一个例子，平面，秩连接格是不允许的。
A note on non-repetitive colourings of planar graphs
Alon等人引入了图的非重复着色的概念。在这里，我们解决一些问题，非重复着色的可平面图。具体来说，我们表明，任何outerplanar地图的脸可以不重复着色使用最多五种颜色。我们还给出了一些下界的颜色所需的数量的非重复着色的顶点的外平面和可平面图。
Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph of maximum degree $4$ obtained as a subgraph of the medial graph of any bipartite plane graph is $3$-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence represen…
Note on robust coloring of planar graphs
本文研究平面图的稳健色数，通过构造一个满足 $χ_1(G) = 3$ 的无限平面图族，解决了 Bacsó 等人近期提出的关于稳健染色的一个问题。
A Note on Odd Colorings of 1-Planar Graphs
本文研究1-平面图的奇数着色问题，证明每个1-平面图都可进行奇数23-着色。
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.
The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\ge 1$, we have $cr_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right)cr(G)$. This bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. Some of the results extend to th…
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
A note on cycles in cyclically $4$-edge-connected cubic planar graphs
研究通过删除两个相邻顶点得到的图H，证明当H的周长至少为k时，存在长度在k到3k/2之间的循环，并展示线图G中可避免特定顶点的循环长度。
Let $F$ be a (possibly improper) edge-coloring of a graph $G$; a vertex coloring of $G$ is \emph{adapted to} $F$ if no color appears at the same time on an edge and on its two endpoints. If for some integer $k$, a graph $G$ is such that given any list assignment $L$ to the vertices of $G$, with $|L(v)| \ge k$ for all $v$, and any edge-coloring $F$ of $G$, $G$ admits a coloring $c$ adapted to $F$ where $c(v) \in L(v)$ for all $v$, then $G$ is said to be \emph{adaptably $k$-choosable}. A {\em $(k,d)$-list assignment} for a graph $G$ is a map that assigns to each vertex $v$ a list $L(v)$ of at le…
A note on irreducible maps with several boundaries
本文导出了具有多个边界的d-不可约二部平面映射的生成函数的一个公式，即具有多个控制度的标记面。它扩展了一个公式，由于Collet和Fusy的情况下，任意（不一定不可约）二分平面映射，我们恢复采取d=0。作为应用，我们得到了每个允许度的面数为给定个数的d-不可约二部平面映射的个数的表达式。对于无重边（d=2）的映射、4-不可约映射和围长至少为6（d=4）的映射，给出了非常明确的表达式.我们的推导是基于一个树的解释遇到的各种生成函数。
Let $Oct_{1}^{+}$ and $Oct_{2}^{+}$ be the planar and non-planar graphs that obtained from the Octahedron by 3-splitting a vertex respectively. For $Oct_{1}^{+}$, we prove that a 4-connected graph is $Oct_{1}^{+}$-free if and only if it is $C_{6}^{2}$, $C_{2k+1}^{2}$ $(k \geq 2)$ or it is obtained from $C_{5}^{2}$ by repeatedly 4-splitting vertices. We also show that a planar graph is $Oct_{1}^{+}$-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from $\{K_{1}, K_{2} ,K_{3}\} \cup \mathscr{K} \cup \{Oct,L_{5} \}$, where $\mathscr{K}$ is the set of graphs obtai…
A note on planar partial 3-trees
它隐含地遵循的工作[Colbourn，埃尔-马拉：在两个对偶类的平面图。离散数学80（1）：21-40（1990）]，每个平面部分3-树是平面3-树的子图。这一事实已经使得能够证明一对夫妇的平面部分3-树的基础平面3-树完成的结构上的归纳的结果。我们提供了一个明确的证明这一观察，并加强它表明，可以保持输入图的平面图不变。
A note on quasi-transitive graphs quasi-isometric to planar (Cayley) graphs
本文证明拟等距于平面图的拟传递图可升级为平面 Cayley 图，回答了 Esperet--Giocanti 和 Hamann 的开放问题。
Neumann-Lara and Škrekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.
We present a family of 3-connected cubic planar Hamiltonian graphs with an exponential number of steps required by Thomason's algorithm. The base of the exponent is approximately $1.1812...$, which exceeds previous results in the area.
A note on Fiedler value of classes with sublinear separators
一类图$\mathcal C$的$n$阶Fiedler值是具有$n$顶点的图$G\in\mathcal C$的最大第二特征值$λ_2(G)$.在这篇文章中，我们将这个值与浅子图联系起来，作为推论，我们确定了一些小闭图类（包括可平面图类）的$n$ -th Fiedler值的正确顺序。
For a planar graph $G$, Borodin stated that $G$ is $(Δ+1)$-edge-choosable if $Δ\geq9$ and later Bonamy showed that $G$ is $9$-edge-choosable if $Δ=8$. At the same time, Borodin et al. proved that $G$ is $Δ$-edge-choosable if $Δ\geq12$. In the paper, we extend these results to $K_5$-minor free graphs.
We call a (not necessarily planar) embedding of a graph $G$ in the plane \emph{sequential} if its vertices lie in $\mathbb Z^2$ and the line segments between adjacent vertices contain no interior integer points. In this note, we prove (i) a graph $G$ has a sequential embedding if and only if $G$ is 4-colorable, and (ii) if $G$ is planar, then $G$ has a sequential planar embedding.
We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex planar graph (vertices are drawn as the given points); this improves the previous best constant $1.235$ by Kurowski (2004). Our second main result is based on exhaustive computer search: We show that no set of 11 points exists, on which all planar 11-vertex graphs can be simultaneously drawn plane straight-line. This strengthens the result by Cardinal, Hoff…
A k-regular planar graph G is nearly Platonic when all faces but one are of the same degree while the remaining face is of a different degree. We show that no such graphs with connectivity one can exist. This complements a recent result by Keith, Froncek, and Kreher on non-existence of 2-connected nearly Platonic graphs.
A note on the structure of locally finite planar quasi-transitive graphs
本研究基于Hamann的工作，证明了3-连通局部有限平面拟传递图的典范树分解结构定理，给出了Hamann等人2022年相关结果的新证明，是拟传递平面图的Droms型结构定理。
Notes on Reeb graphs of real algebraic functions which may not be planar
本文研究无法实现为平面图的 Reeb 图，探讨从这类有限图构造实代数函数及相应流形的问题，拓展了 Sharko 提出的重构研究。
These are lecture notes of the 48th Saint-Flour summer school, July 2018, on the topic of planar maps, random walks and the circle packing theorem.
For a fixed planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In the case when $H$ is a cycle, the asymptotic value of $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_m)$ is currently known for $m\in\{3,4,5,6,8\}$. In this note, we extend this list by establishing $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{10})\sim(n/5)^5$ and $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{12})\sim(n/6)^6$. We prove this by answering the following question for $m\in\{5,6\}$, which is interesting in its own right: which …
Note on Min-k-Planar Drawings of Graphs
本文研究图的 min-k-平面绘制中一般绘制与简单绘制的差异，证明存在具有 min-2-平面绘制或避免相邻边交叉的 min-3-平面绘制的图，但对任意大的固定 k 均不存在简单 min-k-平面绘制。
In this note we determine the set of expansions such that a partial cube is planar if and only if it arises by a sequence of such expansions from a single vertex. This corrects a result of Peterin.
Region crossing change on planar trivalent graphs
本文以平面三价图为对象，考察区域穿越变换所产生的作用。
In this note, we discuss planar lattices generated by their atoms. We prove that if $L$ is a planar lattice generated by $n$ atoms, then both the left and the right boundaries of $L$ have at most $n+1$ elements. On the other hand, $L$ can be arbitrarily large. For every $k > 1$, we construct a planar lattice $L$ generated by $4$ atoms such that $L$ has more than $k$ elements.
A note on the unit distance problem for planar configurations with Q-independent direction set
令$T(n)$表示在欧几里得平面$\mathbb{R}^2$中的一组$n$点可以确定的单位距离的最大数目，其中附加条件是由配置确定的不同单位长度方向必须是$\mathbb{Q}$独立的。这与鄂尔多斯单位距离问题有关，但在方向集上有一个简化的附加假设，该假设“一般地”成立。我们证明了$T(n+1)-T(n)$是$n$的汉明重量，即，$n$的二进制展开式中非零二进制系数的数目，并明确地找到$T(n)$的公式。特别地，$T(n)$是$Θ(n log(n))$。此外，我们描述了一个过程，以构建一个集的$n$点在平面上与$\mathbb{Q}$ -独立的单位长度方向集，精确地达到$T(n)$单位距离。在这样做的过程中，我们表明$T(n)$也是相同的最大数量的边缘的顶点的子集的大小$n$确定的可数无限格$\mathbb{Z}^{\infty}$或无限超立方体图$\{0,1\}^{\infty}$。确定T（n）的问题可以看作是一类填充问题或等周问题。
The Laplacian on planar graphs and graphs on surfaces
这些是2011年11月在哈佛举行的数学会议的讲义。我们讨论的拓扑，概率和组合方面的拉普拉斯算子上的一个图嵌入在一个表面上。主要讨论三个问题：（1）对于“圆形”平面网络，Colin de Verdière对Dirichlet-to-Neumann算子的刻画;（2）与随机生成树模型的联系;（3）环形和环面上拉普拉斯算子的特征多项式。
In recent work of Cachazo, Guevara, Mizera and the author, a generalization of the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ was introduced as an integral over the moduli space of $n$ points in $\mathbb{CP}^{k-1}$, with value a sum of certain rational functions on the kinematic space $\mathcal{K}_{k,n}$. It was shown there for $m^{(3)}(\mathbb{I}_6,\mathbb{I}_6)$ and later by Cachazo and Rojas that collections of poles appearing in $m^{(3)}(\mathbb{I}_7,\mathbb{I}_7)$ are compatible exactly when they are dual to collections of rays which generate the maximal faces of …
Note on the treewidth of graphs excluding a disjoint union of cycles as a minor
本文研究了不包含离散环联合作为小世界的图的树宽界限，证明了其与图顶点数和环数的关系，并得出最优结果。
These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly using pure spinor methods but not restricted to it) in a context that could be of interest to the combinatorics community. In particular, I focused on the appearance of {\it planar binary trees} in scattering amplitudes and presented some curious identities obeyed by related objects, some of which are known to be true only via explicit examples.
$k$-quasi planar graphs
一个拓扑图是\emph{$k$-quasi-planar}，如果它不包含$k$成对交叉边。一个拓扑图是\emph{simple}，如果它的每对边相交最多一次（在一个顶点或在它们的交点）。1996年，Pach，Shahrokhi，and Szegedy证明了每个$n$ -顶点简单$k$ -拟平面图至多包含$O(n(\log n)^{2k-4})$边.最近Fox和Pach \cite{fox}将该上限（对于大$k$）改进为$n(\log n)^{O(\log k)}$。本文证明了所有这类图至多包含$(n\log^2n)2^{α^{c_k}(n)}$边，其中$α(n)$表示逆阿克曼函数，$c_k$是一个只依赖于$k$的常数.
Conformal invariance of lattice models
这些课堂讲稿提供了一个（几乎）自足的帐户共形不变的平面临界伊辛和FK伊辛模型。他们提出了离散全纯函数理论及其在平面统计物理学中的应用（更确切地说是在费米观测量的收敛中）。收敛到SLE的简要讨论。其中包括许多开放性问题。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。