Journal of Graph Theory最新文献_第10页

Ramsey numbers upon vertex deletion 顶点删除后的拉姆齐数

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-03-18 DOI: 10.1002/jgt.23093

Yuval Wigderson

Given a graph <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math>, its Ramsey number <math> <semantics> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> <annotation> $r(G)$</annotation> </semantics></math> is the minimum <math> <semantics> <mrow> <mi>N</mi> </mrow> <annotation> $N$</annotation> </semantics></math> so that every two-coloring of <math> <semantics> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi>N</mi> </msub> <mo>)</mo> </mrow> </mrow> <annotation> $E({K}_{N})$</annotation> </semantics></math> contains a monochromatic copy of <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math>. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math>, the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs <math> <semantics> <mrow> <mo>{</mo> <msub> <mi>G</mi> <mi>n</mi> </msub> <mo>}</mo> </mrow> <annotation> ${{G}_{n}}$</annotation> </semantics></math> so that in any Ramsey coloring for <math> <semantics> <mrow> <msub> <mi>G</mi>

康伦、福克斯和苏达科夫猜想，如果从一个图中删除一个顶点，拉姆齐数最多只能以一个常数因子变化。我们推翻了这一猜想，展示了一个无限图族，从每个图中删除一个顶点都会使拉姆齐数减少一个超常数因子。这一结果的一个结果如下。存在这样一个图族：在任何拉姆齐着色（即没有单色副本的顶点上的一个小群的着色）中，其中一个颜色类的密度为 .

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引用次数: 0

Flip distance and triangulations of a polyhedron 多面体的翻转距离和三角形

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-03-18 DOI: 10.1002/jgt.23096

Zili Wang

It is known that the flip distance between two triangulations of a convex polygon is related to the smallest number of tetrahedra in the triangulation of some polyhedron. The latter was used to compute the diameter of the flip graph of convex polygons with a large number of vertices. However, it is yet unknown whether the flip distance and this smallest number of tetrahedra are always the same or even close. In this work, we find examples to show that the ratio between these two numbers can be arbitrarily close to $� � � \frac{�}{3} � � �$ . We also propose two conjectures in the end, one about this ratio, and the other may have some implications on when two triangulations can achieve maximal distance.

众所周知，凸多边形的两个三角形之间的翻转距离与某个多面体的三角形中的最小四面体数有关。后者被用来计算具有大量顶点的凸多边形的翻转图形直径。然而，翻转距离和这个最小的四面体数目是否总是相同甚至接近，目前还不得而知。在这项工作中，我们找到了一些例子来证明这两个数字之间的比值可以任意地接近于。最后，我们还提出了两个猜想，一个是关于这个比值的，另一个可能对两个三角形何时能达到最大距离有一些影响。

引用次数: 0

Finding triangle-free 2-factors in general graphs 在一般图形中寻找无三角形的 2 因子

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-03-08 DOI: 10.1002/jgt.23089

David Hartvigsen

A 2-factor in a graph $� � � G � � �$ is a subset of edges $� � � M � � �$ such that every node of $� � � G � � �$ is incident with exactly two edges of $� � � M � � �$ . Many results are known concerning 2-factors including a polynomial-time algorithm for finding 2-factors and a characterization of those graphs that have a 2-factor. The problem of finding a 2-factor in a graph is a relaxation of the NP-hard problem of finding a Hamilton cycle. A stronger relaxation is the problem of finding a triangle-free 2-factor, that is, a 2-factor whose edges induce no cycle of length 3. In this paper, we present a polynomial-time algorithm for the problem of finding a triangle-free 2-factor as well as a characterization of the graphs that have such a 2-factor and related min–max and augmenting path theorems.

图 G$G$ 中的 2 因子是边 M$M$ 的一个子集，使得 G$G$ 的每个节点都正好与 M$M$ 的两条边相连。有关 2 因子的许多结果已经为人所知，其中包括寻找 2 因子的多项式时间算法和具有 2 因子的图的特征描述。在图中寻找 2 因子的问题是寻找汉密尔顿循环这一 NP 难问题的松弛。本文提出了寻找无三角形 2 因子问题的多项式时间算法，以及具有这种 2 因子的图的特征和相关的最小-最大和增强路径定理。

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引用次数: 0

Orientations of graphs with maximum Wiener index 具有最大维纳指数的图形方向

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-03-03 DOI: 10.1002/jgt.23090

Zhenzhen Li, Baoyindureng Wu

In this paper, we study the Wiener index of the orientation of trees and theta-graphs. An orientation of a tree is called no-zig-zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree <math> <semantics> <mrow> <mi>T</mi> </mrow> <annotation> $T$</annotation> </semantics></math> achieving the maximum Wiener index is no-zig-zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta-graph <math> <semantics> <mrow> <msub> <mi>Θ</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> </mrow> <annotation> ${{rm{Theta }}}_{a,b,c}$</annotation> </semantics></math> with <math> <semantics> <mrow> <mi>a</mi> <mo>≥</mo> <mi>b</mi> <mo>≥</mo> <mi>c</mi> </mrow> <annotation> $age bge c$</annotation> </semantics></math> and <math> <semantics> <mrow> <mi>b</mi> <mo>></mo> <mn>1</mn> </mrow> <annotation> $bgt 1$</annotation> </semantics></math>, the maximum Wiener index is achieved by the one in which the union of the paths between <math> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <annotation> ${u}_{1}$</annotation> </semantics></math> and <math> <semantics> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <annotation> ${u}_{2}$</annotation> </semantics></math> forms a directed cycle of length <math> <semantics> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo>

本文研究了树和θ图的方向的维纳指数。如果树的一个方向上不存在边改变方向两次的子路径，则称为无之字形方向。Knor、Škrekovski 和 Tepeh 猜想，达到最大维纳指数的树的每个方向都是无之字形。我们通过构建一个反例推翻了这一猜想。Knor、Škrekovski 和 Tepeh 猜想，在有和的θ图的所有方向中，达到最大维纳指数的方向是和之间的路径联合形成长度为和的有向循环的方向，其中和是阶数为 3 的顶点。

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引用次数: 0

Sharp lower bounds for the number of maximum matchings in bipartite multigraphs 双方格多图中最大匹配数的锐下限

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-03-03 DOI: 10.1002/jgt.23080

Alexandr V. Kostochka, Douglas B. West, Zimu Xiang

We study the minimum number of maximum matchings in a bipartite multigraph <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> with parts <math> <semantics> <mrow> <mi>X</mi> </mrow> <annotation> $X$</annotation> </semantics></math> and <math> <semantics> <mrow> <mi>Y</mi> </mrow> <annotation> $Y$</annotation> </semantics></math> under various conditions, refining the well-known lower bound due to M. Hall. When <math> <semantics> <mrow> <mo>∣</mo> <mi>X</mi> <mo>∣</mo> <mo>=</mo> <mi>n</mi> </mrow> <annotation> $| X| =n$</annotation> </semantics></math>, every vertex in <math> <semantics> <mrow> <mi>X</mi> </mrow> <annotation> $X$</annotation> </semantics></math> has degree at least <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>, and every vertex in <math> <semantics> <mrow> <mi>X</mi> </mrow> <annotation> $X$</annotation> </semantics></math> has at least <math> <semantics> <mrow> <mi>r</mi> </mrow> <annotation> $r$</annotation> </semantics></math> distinct neighbors, the minimum is <math> <semantics> <mrow> <mi>r</mi> <mo>!</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $r!(k-r+1)$</annotation> </semantics></math> when <math> <semantics> <mrow> <mi>n</mi> <mo>≥</mo> <mi>r</mi> </mrow> <annotation> $nge r$</annotation> </semantics></math> and is <math> <semantics> <mrow> <mrow> <mo>[</mo> <mrow> <mi>r</mi>

我们研究了具有部分和的双方形多图在各种条件下的最大匹配数的最小值，完善了霍尔（M. Hall）提出的著名下界。当，中的每个顶点都至少有度，且每个顶点都至少有不同的邻居时，最小匹配数为，且为。当每个顶点至少有两个邻居且时，最小值为，其中。我们还确定了其他几种情况下最大匹配数的最小值。我们提供了多种锐度构造。

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引用次数: 0

Extremal spectral results of planar graphs without vertex-disjoint cycles 无顶点相交循环平面图的极谱结果

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-02-28 DOI: 10.1002/jgt.23084

Longfei Fang, Huiqiu Lin, Yongtang Shi

Given a planar graph family <math> <semantics> <mrow> <mi>F</mi> </mrow> <annotation> ${rm{ {mathcal F} }}$</annotation> </semantics></math>, let <math> <semantics> <mrow> <mi>e</mi> <msub> <mi>x</mi> <mi>P</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo>)</mo> </mrow> </mrow> <annotation> $e{x}_{{mathscr{P}}}(n,{mathscr{F}})$</annotation> </semantics></math> and <math> <semantics> <mrow> <mi>s</mi> <mi>p</mi> <mi>e</mi> <msub> <mi>x</mi> <mi>P</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mi>F</mi> <mo>)</mo> </mrow> </mrow> <annotation> $spe{x}_{{mathscr{P}}}(n,{mathscr{F}})$</annotation> </semantics></math> be the maximum size and maximum spectral radius over all <math> <semantics> <mrow> <mi>n</mi> </mrow> <annotation> $n$</annotation> </semantics></math>-vertex <math> <semantics> <mrow> <mi>F</mi> </mrow> <annotation> ${rm{ {mathcal F} }}$</annotation> </semantics></math>-free planar graphs, respectively. Let <math> <semantics> <mrow> <mi>t</mi> <msub> <mi>C</mi> <mi>ℓ</mi> </msub> </mrow> <annotation> $t{C}_{ell }$</annotation> </semantics></math> be the disjoint union of <math> <semantics> <mrow> <mi>t</mi> </mrow> <annotation> $t$</annotation> </semantics></math> copies of <math> <semantics> <mrow> <mi>ℓ</mi> </mrow> <annotation> $ell $</annotation> </semantics></math>-cycles, and <math> <semantics> <mrow> <mi>t</mi> <mi>C</mi> </mrow> <annotation> $t{mathscr{C}}$</annotation> </semantics></math> be the family of <math> <semantics>

给定一个平面图族，设和分别是所有无顶点平面图的最大尺寸和最大谱半径。设为 - 循环的副本的不相邻联盟，且为无长度限制的顶点不相邻循环族。Tait 和 Tobin 确定，是所有阶数足够大的平面图中的极值谱图，这意味着和的极值图都是。在本文中，我们首先确定了和，并描述了对于、和足够大的唯一极值图。其次，我们得到了和的精确值，从而解决了 Li 对的猜想。

{"title":"Extremal spectral results of planar graphs without vertex-disjoint cycles","authors":"Longfei Fang, Huiqiu Lin, Yongtang Shi","doi":"10.1002/jgt.23084","DOIUrl":"10.1002/jgt.23084","url":null,"abstract":"Given a planar graph family <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}$</annotation>\u0000 </semantics></math>, let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>e</mi>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>P</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>F</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $e{x}_{{mathscr{P}}}(n,{mathscr{F}})$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mi>p</mi>\u0000 <mi>e</mi>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>P</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>F</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $spe{x}_{{mathscr{P}}}(n,{mathscr{F}})$</annotation>\u0000 </semantics></math> be the maximum size and maximum spectral radius over all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal F} }}$</annotation>\u0000 </semantics></math>-free planar graphs, respectively. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mi>ℓ</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $t{C}_{ell }$</annotation>\u0000 </semantics></math> be the disjoint union of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math> copies of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 </mrow>\u0000 <annotation> $ell $</annotation>\u0000 </semantics></math>-cycles, and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation> $t{mathscr{C}}$</annotation>\u0000 </semantics></math> be the family of <math>\u0000 <semantics>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 3","pages":"496-524"},"PeriodicalIF":0.9,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Threshold for stability of weak saturation 弱饱和稳定性阈值

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-02-23 DOI: 10.1002/jgt.23079

Mohammadreza Bidgoli, Ali Mohammadian, Behruz Tayfeh-Rezaie, Maksim Zhukovskii

We study the weak <math> <semantics> <mrow> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> <annotation> ${K}_{s}$</annotation> </semantics></math>-saturation number of the Erdős–Rényi random graph <math> <semantics> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> ${mathbb{G}}(n,p)$</annotation> </semantics></math>, denoted by <math> <semantics> <mrow> <mtext>wsat</mtext> <mrow> <mo>(</mo> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $text{wsat}({mathbb{G}}(n,p),{K}_{s})$</annotation> </semantics></math>, where <math> <semantics> <mrow> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> <annotation> ${K}_{s}$</annotation> </semantics></math> is the complete graph on <math> <semantics> <mrow> <mi>s</mi> </mrow> <annotation> $s$</annotation> </semantics></math> vertices. In 2017, Korándi and Sudakov proved that the weak <math> <semantics> <mrow> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> <annotation> ${K}_{s}$</annotation> </semantics></math>-sat

我们研究厄尔多斯-雷尼随机图的弱饱和数，用，表示，其中是顶点上的完整图。2017 年，Korándi 和 Sudakov 证明了的弱饱和数是稳定的，即在以恒定概率移除边后，它保持不变。在本文中，我们证明了这一稳定性存在一个阈值，并给出了阈值的上界和下界。这推广了 Korándi 和 Sudakov 的结果。本文还给出了一般的上界。

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引用次数: 0

The structure of digraphs with excess one 多一数图的结构

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-02-19 DOI: 10.1002/jgt.23082

James Tuite

A digraph <math> <semantics> <mrow> <mi>G</mi> </mrow> <annotation> $G$</annotation> </semantics></math> is <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-geodetic if for any (not necessarily distinct) vertices <math> <semantics> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> <annotation> $u,v$</annotation> </semantics></math> there is at most one directed walk from <math> <semantics> <mrow> <mi>u</mi> </mrow> <annotation> $u$</annotation> </semantics></math> to <math> <semantics> <mrow> <mi>v</mi> </mrow> <annotation> $v$</annotation> </semantics></math> with length not exceeding <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>. The order of a <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-geodetic digraph with minimum out-degree <math> <semantics> <mrow> <mi>d</mi> </mrow> <annotation> $d$</annotation> </semantics></math> is bounded below by the directed Moore bound <math> <semantics> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mo>+</mo> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>⋯</mi> <mo>+</mo> <msup> <mi>d</mi> <mi>k</mi> </msup> </mrow> <annotation> $M(d,

如果对于任何（不一定不同的）顶点，从到的有向行走最多只有一次，且长度不超过 .具有最小外度的有向有序数图的阶数受有向摩尔约束的约束。摩尔约束只有在琐碎的情况下才能满足，因此我们有兴趣寻找具有外度和最小可能阶数的-大地数字图，其中阶数是数字图的过量。米勒（Miller）、米雷特（Miret）和西拉森（Sillasen）最近排除了在和时存在过量为 1 的图的可能性。我们猜想，在本文中，我们将研究这一猜想的最小反例的结构。我们严格限制了离群函数的可能结构，并证明了某些阶数为三且多余度为一的图的不存在性，以及米勒等人的分析所留下的开放情况和的关闭情况。我们进一步证明了不存在多余度为一的渐开线图，也就是说，任何此类图的离群函数必须包含一个长度为的循环。

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引用次数: 0

Self-avoiding walks and polygons on hyperbolic graphs 双曲图上的自避走和多边形

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-02-19 DOI: 10.1002/jgt.23087

Christoforos Panagiotis

We prove that for the <math> <semantics> <mrow> <mi>d</mi> </mrow> <annotation> $d$</annotation> </semantics></math>-regular tessellations of the hyperbolic plane by <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>-gons, there are exponentially more self-avoiding walks of length <math> <semantics> <mrow> <mi>n</mi> </mrow> <annotation> $n$</annotation> </semantics></math> than there are self-avoiding polygons of length <math> <semantics> <mrow> <mi>n</mi> </mrow> <annotation> $n$</annotation> </semantics></math>. We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed <math> <semantics> <mrow> <mi>k</mi> </mrow> <annotation> $k$</annotation> </semantics></math>, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion <math> <semantics> <mrow> <mi>d</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>O</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>∕</mo> <mi>d</mi> </mrow> <mo>)</mo> </mrow> </mrow> <annotation> $d-1-O(1unicode{x02215}d)$</annotation> </semantics></math> as <math> <semantics> <mrow> <mi>d</mi> <mo>→</mo> <mi>∞</mi> </mrow> <annotation> $dto infty $</annotation> </semantics></math>; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length <math> <semantics> <mrow> <

我们证明，对于由 k$k$ 图案构成的双曲面 d$d$ 不规则方格网，长度为 n$n$ 的自避走比长度为 n$n$ 的自避多边形多出指数级。然后，我们证明这一特性意味着自避让行走是弹道的，即使在任意顶点传递图上也是如此。此外，对于每一个固定的 k$k$，我们证明自避让行走的连接常数满足 d→∞$dto infty $ 时的渐近展开 d-1-O(1∕d)$d-1-O(1unicode{x02215}d)$ ；另一方面，自避让多边形的连接常数仍然是有界的。最后，我们证明了除两个棋盘格外，长度为 n$n$ 的自回避步行的数量与它们的连接常数的 n$n$ 次幂相当。其中一些结果是马德拉斯和吴先前针对双曲面中除有限多个规则方格之外的所有方格得到的。

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引用次数: 0

Polyhedra without cubic vertices are prism-hamiltonian 无立方顶点的多面体是棱-汉密尔顿多面体

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-02-19 DOI: 10.1002/jgt.23078

Simon Špacapan

The prism over a graph $� � � G � � �$ is the Cartesian product of $� � � G � � �$ with the complete graph on two vertices. A graph $� � � G � � �$ is prism-hamiltonian if the prism over $� � � G � � �$ is hamiltonian. We prove that every polyhedral graph (i.e., 3-connected planar graph) of minimum degree at least four is prism-hamiltonian.

图 G$G$ 上的棱是 G$G$ 与两个顶点上的完整图的笛卡尔积。如果 G$G$ 上的棱是哈密顿的，那么图 G$G$ 就是棱哈密顿的。我们证明了每一个最小阶数至少为四的多面体图（即三连平面图）都是棱-哈密顿图。

{"title":"Polyhedra without cubic vertices are prism-hamiltonian","authors":"Simon Špacapan","doi":"10.1002/jgt.23078","DOIUrl":"10.1002/jgt.23078","url":null,"abstract":"The prism over a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is the Cartesian product of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with the complete graph on two vertices. A graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is prism-hamiltonian if the prism over <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is hamiltonian. We prove that every polyhedral graph (i.e., 3-connected planar graph) of minimum degree at least four is prism-hamiltonian.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 2","pages":"380-406"},"PeriodicalIF":0.9,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23078","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139953683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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