Journal of Graph Theory最新文献_第5页

On vertex-transitive graphs with a unique hamiltonian cycle 关于具有唯一哈密顿循环的顶点变换图

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-08-22 DOI: 10.1002/jgt.23166

Babak Miraftab, Dave Witte Morris

A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group $� � � �_{F � � n �} �$ has a Cayley graph of degree $� � � 2 � � n � � + � � 2 �$ that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of $� � � �_{F � � n �} �$ are outerplanar.

如果一个图具有唯一的哈密顿循环，那么这个图就是唯一哈密顿图。为了将这一概念自然扩展到无限图，我们找到了所有具有有限多个末端的唯一哈密顿顶点传递图，并讨论了一些具有无限多个末端的例子。特别是，我们证明了每个非阿贝尔自由群都有一个具有唯一哈密顿圆的阶数 Cayley 图。(此外，我们还证明了这些 Cayley 图是外平面的。

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引用次数: 0

Nonisomorphic two-dimensional algebraically defined graphs over R

R

R R 上的非同构二维代数定义图

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-08-19 DOI: 10.1002/jgt.23161

Brian G. Kronenthal, Joe Miller, Alex Nash, Jacob Roeder, Hani Samamah, Tony W. H. Wong

For <math> <mrow> <mi>f</mi> <mo>:</mo> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>→</mo> <mi>R</mi> </mrow></math>, let <math> <mrow> <msub> <mi>Γ</mi> <mi>R</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow></math> be a two-dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of <math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> </mrow></math> and two vertices <math> <mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow></math> and <math> <mrow> <mrow> <mo>[</mo> <mrow> <mi>x</mi> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> <mo>]</mo> </mrow> </mrow></math> are adjacent if and only if <math> <mrow> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow></

对于 f : R 2 → R ，设 Γ R ( f ) 是一个二维代数定义图，即一个二元图，其中每个部分集都是 R 2 的副本，并且当且仅当 a 2 + x 2 = f ( a , x ) 时，两个顶点 ( a , a 2 ) 和 [ x , x 2 ] 相邻。已知 Γ R ( X Y ) 的周长为 6，可以扩展为经典实射影平面的点线入射图。然而，是否存在 f ∈ R [ X , Y ] 使得 Γ R ( f ) 的周长为 6 并且与 Γ R ( X Y ) 非同构的情况，这还是个未知数。本文肯定地回答了这个问题，从而提供了一个非经典实射影平面的构造。本文还研究了二元函数 f 族的Γ R ( f ) 的直径和周长。

{"title":"Nonisomorphic two-dimensional algebraically defined graphs over \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23161:jgt23161-math-0001\" wiley:location=\"equation/jgt23161-math-0001.png\"><mrow><mrow><mi mathvariant=\"double-struck\">R</mi></mrow></mrow></math>","authors":"Brian G. Kronenthal, Joe Miller, Alex Nash, Jacob Roeder, Hani Samamah, Tony W. H. Wong","doi":"10.1002/jgt.23161","DOIUrl":"https://doi.org/10.1002/jgt.23161","url":null,"abstract":"For <math>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>→</mo>\u0000 \u0000 <mi>R</mi>\u0000 </mrow></math>, let <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>Γ</mi>\u0000 \u0000 <mi>R</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>f</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> be a two-dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of <math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow></math> and two vertices <math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>a</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> and <math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mi>x</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>x</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow></math> are adjacent if and only if <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>a</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <msub>\u0000 <mi>x</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>f</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>x</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"50-64"},"PeriodicalIF":0.9,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707817","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

Five-cycle double cover and shortest cycle cover 五周期双覆盖和最短周期覆盖

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-08-11 DOI: 10.1002/jgt.23164

Siyan Liu, Rong-Xia Hao, Rong Luo, Cun-Quan Zhang

The 5-even subgraph cycle double cover conjecture (5-CDC conjecture) asserts that every bridgeless graph has a 5-even subgraph double cover. A shortest even subgraph cover of a graph <math> <mrow> <mi>G</mi> </mrow></math> is a family of even subgraphs which cover all the edges of <math> <mrow> <mi>G</mi> </mrow></math> and the sum of their lengths is minimum. It is conjectured that every bridgeless graph <math> <mrow> <mi>G</mi> </mrow></math> has an even subgraph cover with total length at most <math> <mrow> <mfrac> <mn>21</mn> <mn>15</mn> </mfrac> <mo>∣</mo> <mi>E</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>∣</mo> </mrow></math>. In this paper, we study those two conjectures for weak oddness 2 cubic graphs and present a sufficient condition for such graphs to have a 5-CDC containing a member with many vertices. As a corollary, we show that for every oddness 2 cubic graph <math> <mrow> <mi>G</mi> </mrow></math> satisfying the sufficient condition has a 4-even subgraph <math> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow></math>-cover with total length at most <math> <mrow> <mfrac> <mn>20</mn> <mn>15</mn> </mfrac> <mo>∣</mo> <mi>E</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>∣</mo> <mo>+</mo> <mn>2</mn> </mrow></math>. We also show that every oddness 2 cubic graph <math> <mrow> <mi>G</mi> </mrow></math> with girth at least 30 has a 5-CDC containing a member of length at least <math> <mrow> <mfrac> <mn>9</mn>

5-偶数子图循环双覆盖猜想（5-CDC 猜想）断言每个无桥图都有一个 5-偶数子图双覆盖。一个图的最短偶数子图盖是偶数子图族，它们覆盖了图的所有边，并且它们的长度之和最小。有人猜想，每个无桥图都有一个总长度至多为的偶数子图盖。在本文中，我们研究了弱奇数 2 立方图的这两个猜想，并提出了此类图具有包含一个有多个顶点的成员的 5-CDC 的充分条件。作为推论，我们证明了满足充分条件的每个奇度为 2 的立方图都有一个总长度最多为 .我们还证明，每个周长至少为 30 的奇度为 2 的立方图都有一个包含至少一个长度为的成员的 5-CDC ，因此它有一个总长度至多为的 4-even 子图-覆盖。

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引用次数: 0

Dense circuit graphs and the planar Turán number of a cycle 密集电路图和循环的平面图兰数

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-08-07 DOI: 10.1002/jgt.23165

Ruilin Shi, Zach Walsh, Xingxing Yu

The planar Turán number <math> <mrow> <msub> <mtext>ex</mtext> <mi>P</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>H</mi> </mrow> <mo>)</mo> </mrow> </mrow></math> of a graph <math> <mrow> <mi>H</mi> </mrow></math> is the maximum number of edges in an <math> <mrow> <mi>n</mi> </mrow></math>-vertex planar graph without <math> <mrow> <mi>H</mi> </mrow></math> as a subgraph. Let <math> <mrow> <msub> <mi>C</mi> <mi>k</mi> </msub> </mrow></math> denote the cycle of length <math> <mrow> <mi>k</mi> </mrow></math>. The planar Turán number <math> <mrow> <msub> <mtext>ex</mtext> <mi>P</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <msub> <mi>C</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow></math> is known for <math> <mrow> <mi>k</mi> <mo>≤</mo> <mn>7</mn> </mrow></math>. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Turán numbers. In particular, we prove that there is a constant <math> <mrow> <mi>D</mi> </mrow></math> so that <math> <mrow> <msub> <mtext>ex</mtext>

图的平面图兰数是一个无顶点平面图的最大边数。让表示长度为的循环。平面图兰数是已知的。我们证明了具有一定连通性的密集平面图（称为回路图）包含大的近三角形，并利用这一结果得到了平面图兰数的结果。特别是，我们证明存在一个常数，使得对于所有和。当这一约束严格到常数时，就证明了 Cranston、Lidický、Liu 和 Shantanam 的猜想。

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引用次数: 0

Strong arc decompositions of split digraphs 分裂图的强弧分解

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-08-06 DOI: 10.1002/jgt.23157

Jørgen Bang-Jensen, Yun Wang

A strong arc decomposition of a digraph <math> <mrow> <mi>D</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>,</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> </mrow></math> is a partition of its arc set <math> <mrow> <mi>A</mi> </mrow></math> into two sets <math> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> </mrow></math> such that the digraph <math> <mrow> <msub> <mi>D</mi> <mi>i</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>,</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow></math> is strong for <math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow></math>. Bang-Jensen and Yeo conjectured that there is some <math> <mrow> <mi>K</mi> </mrow></math> such that every <math> <mrow> <mi>K</mi> </mrow></math>-arc-strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2-arc-strong locally semicomplete digraph which is not the square of an even cycle h

一个数图的强弧分解是将其弧集划分为两个集，使得该数图对。Bang-Jensen 和 Yeo 猜想，存在这样一种情况，即每个-弧强的数图都有一个强弧分解。他们还证明，除了在四个顶点上有一个例外，每个 2 弧强半完全数图都有一个强弧分解。Bang-Jensen 和 Huang 将这一结果扩展到局部半完全数图，证明了每一个不是偶数循环的平方的 2 弧强局部半完全数图都具有强弧分解。这意味着每一个 3 弧强局部半完全数图都有一个强弧分解。分裂图是指底层无向图为分裂图的图，即其顶点可划分为一个小群和一个独立集。等价地，分裂数图是任何数图，它可以通过添加新的顶点集和一些与之间的弧从半完整数图中得到。在本文中，我们证明了每一个 3 弧强分裂数图都有一个强弧分解，可以在多项式时间内找到，我们还提供了无穷类没有强弧分解的 2 强分裂数图。我们还提出了一些关于分裂图的未决问题。

{"title":"Strong arc decompositions of split digraphs","authors":"Jørgen Bang-Jensen, Yun Wang","doi":"10.1002/jgt.23157","DOIUrl":"10.1002/jgt.23157","url":null,"abstract":"A strong arc decomposition of a digraph <math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>A</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is a partition of its arc set <math>\u0000 \u0000 <mrow>\u0000 <mi>A</mi>\u0000 </mrow></math> into two sets <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow></math> such that the digraph <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is strong for <math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math>. Bang-Jensen and Yeo conjectured that there is some <math>\u0000 \u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow></math> such that every <math>\u0000 \u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow></math>-arc-strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2-arc-strong locally semicomplete digraph which is not the square of an even cycle h","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"5-26"},"PeriodicalIF":0.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23157","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943301","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}

引用次数: 0

Stability of Rose Window graphs 玫瑰窗图形的稳定性

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-08-05 DOI: 10.1002/jgt.23162

Milad Ahanjideh, István Kovács, Klavdija Kutnar

A graph $� � � � � Γ � � � �$ is said to be stable if for the direct product $� � � � � Γ � � \times � � �_{K � � 2 �} � �, � � Aut � � � (� � � Γ � � \times � � �_{K � � 2 �} � � �) � � � � �$ is isomorphic to $� � � � � Aut � � � (� � Γ � �) � � � \times � � �_{Z � � 2 �} � � � �$ ; otherwise, it is called unstable. An unstable graph is called nontrivially unstable when it is not bipartite and no two vertices have the same neighborhood. Wilson described nine families of unstable Rose Window graphs and conjectured that these contain all nontrivially unstable Rose Window graphs (2008). In this paper we show that the conjecture is true.

如果图的直积与图同构，则称该图为稳定图；反之，则称该图为不稳定图。如果一个不稳定图不是两部分的，且没有两个顶点具有相同的邻域，则称为非两部分不稳定图。Wilson 描述了九个不稳定玫瑰窗图族，并猜想这些族包含所有非难不稳定玫瑰窗图（2008 年）。在本文中，我们证明了猜想的正确性。

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引用次数: 0

Proper edge colorings of planar graphs with rainbow C 4 ${C}_{4}$ -s 具有彩虹 C4 ${C}_{4}$-s 的平面图的适当边着色

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-08-05 DOI: 10.1002/jgt.23163

András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy

We call a proper edge coloring of a graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> a B-coloring if every 4-cycle of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> is colored with four different colors. Let <math> <semantics> <mrow> <mrow> <msub> <mi>q</mi> <mi>B</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${q}_{B}(G)$</annotation> </semantics></math> denote the smallest number of colors needed for a B-coloring of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>. Motivated by earlier papers on B-colorings, here we consider <math> <semantics> <mrow> <mrow> <msub> <mi>q</mi> <mi>B</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${q}_{B}(G)$</annotation> </semantics></math> for planar and outerplanar graphs in terms of the maximum degree <math> <semantics> <mrow> <mrow> <mi>Δ</mi> <mo>=</mo> <mi>Δ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${rm{Delta }}={rm{Delta }}(G)$</annotation> </sem

如果图的每个 4 循环都用四种不同的颜色着色，我们就称该图的适当边着色为 B 着色。让表示 B 染色所需的最小颜色数。受早先关于 B 染色的论文的启发，我们在此考虑平面图和外平面图的最大度。我们证明，对于平面图、双方形平面图以及具有 .我们猜想，在足够大的情况下，对于平面图 , 和外平面图 .

{"title":"Proper edge colorings of planar graphs with rainbow \u0000 \u0000 \u0000 \u0000 \u0000 C\u0000 4\u0000 \u0000 \u0000 \u0000 ${C}_{4}$\u0000 -s","authors":"András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy","doi":"10.1002/jgt.23163","DOIUrl":"10.1002/jgt.23163","url":null,"abstract":"We call a proper edge coloring of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> a B-coloring if every 4-cycle of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is colored with four different colors. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>q</mi>\u0000 \u0000 <mi>B</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${q}_{B}(G)$</annotation>\u0000 </semantics></math> denote the smallest number of colors needed for a B-coloring of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. Motivated by earlier papers on B-colorings, here we consider <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>q</mi>\u0000 \u0000 <mi>B</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${q}_{B}(G)$</annotation>\u0000 </semantics></math> for planar and outerplanar graphs in terms of the maximum degree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>Δ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Delta }}={rm{Delta }}(G)$</annotation>\u0000 </sem","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"833-846"},"PeriodicalIF":0.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943351","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

Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs” 对 "在随机几何图中嵌入平衡生成树的锐阈值 "的更正

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-31 DOI: 10.1002/jgt.23160

A. Espuny Díaz, L. Lichev, D. Mitsche, and A. Wesolek, Sharp threshold for embedding balanced spanning trees in random geometric graphs, J. Graph Theory. 107 (2024), 107–125. https://doi.org/10.1002/jgt.23106

In the “ACKNOWLEDGMENTS” section, the text “The research leading to these results has been supported by the Carl-Zeiss-Foundation (Alberto Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002, and by Fondecyt grant 1220174 (Dieter Mitsche) and by the Vanier Scholarship Program (Alexandra Wesolek).” was incorrect.

This should have read: “The research leading to these results has been supported by the Carl-Zeiss-Foundation and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project no. 447645533 (A. Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002 and by Fondecyt grant 1220174 (D. Mitsche) and by the Vanier Scholarship Program (A. Wesolek).”

The new Funding Information should read as follows:

Carl-Zeiss-Foundation.

Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project no. 447645533.

Grant GrHyDy ANR-20-CE40-0002.

Fondecyt, Grant/Award Number: 1220174.

The Vanier Scholarship Program.

We apologize for this error.

A.Espuny Díaz, L. Lichev, D. Mitsche, and A. Wesolek, Sharp threshold for embedding balanced spanning trees in random geometric graphs, J. Graph Theory.107 (2024)，107-125。https://doi.org/10.1002/jgt.23106In "致谢 "部分，"这些成果的研究得到了卡尔-蔡司基金会（Alberto Espuny Díaz）、GrHyDy ANR-20-CE40-0002 补助金、Fondecyt 1220174 补助金（Dieter Mitsche）和 Vanier 奖学金项目（Alexandra Wesolek）的支持。"有误。应为 "这些成果的研究得到了卡尔-蔡司基金会（Alberto Espuny Díaz）、GrHyDy ANR-20-CE40-0002 补助金、Fondecyt 1220174 补助金（Dieter Mitsche）和 Vanier 奖学金项目（Alexandra Wesolek）的支持："产生这些成果的研究得到了卡尔-蔡司基金会和德国研究基金会（DFG）的支持，项目编号为 447645533 (A. Espuny Díaz)，资助金为 GrHyDy ANR-20-CE40-0002 和 Fondecyt 资助金 1220174 (D. Mitsche) 以及 Vanier 奖学金计划 (A. Wesolek)。新的资助信息应为：Carl-Zeiss-Foundation.Deutsche Forschungsgemeinschaft (DFG，德国研究基金会)，项目编号：447645533.GrHyDy ANR-20-CE40-0002.Fondecyt, Grant/Award Number: 1220174.The Vanier Scholarship Program.我们对这一错误表示歉意。

{"title":"Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”","authors":"","doi":"10.1002/jgt.23160","DOIUrl":"10.1002/jgt.23160","url":null,"abstract":"A. Espuny Díaz, L. Lichev, D. Mitsche, and A. Wesolek, Sharp threshold for embedding balanced spanning trees in random geometric graphs, J. Graph Theory. 107 (2024), 107–125. https://doi.org/10.1002/jgt.23106In the “ACKNOWLEDGMENTS” section, the text “The research leading to these results has been supported by the Carl-Zeiss-Foundation (Alberto Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002, and by Fondecyt grant 1220174 (Dieter Mitsche) and by the Vanier Scholarship Program (Alexandra Wesolek).” was incorrect.This should have read: “The research leading to these results has been supported by the Carl-Zeiss-Foundation and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project no. 447645533 (A. Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002 and by Fondecyt grant 1220174 (D. Mitsche) and by the Vanier Scholarship Program (A. Wesolek).”The new Funding Information should read as follows:Carl-Zeiss-Foundation.Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project no. 447645533.Grant GrHyDy ANR-20-CE40-0002.Fondecyt, Grant/Award Number: 1220174.The Vanier Scholarship Program.We apologize for this error.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"847"},"PeriodicalIF":0.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866312","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}

引用次数: 0

On the minimum number of arcs in 4-dicritical oriented graphs 论四临界定向图中的最小弧数

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-29 DOI: 10.1002/jgt.23159

Frédéric Havet, Lucas Picasarri-Arrieta, Clément Rambaud

The dichromatic number <math> <semantics> <mrow> <mrow> <mover> <mi>χ</mi> <mo>→</mo> </mover> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $overrightarrow{chi }(D)$</annotation> </semantics></math> of a digraph <math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> <annotation> $D$</annotation> </semantics></math> is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph <math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> <annotation> $D$</annotation> </semantics></math> is <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> <annotation> $k$</annotation> </semantics></math>-dicritical if <math> <semantics> <mrow> <mrow> <mover> <mi>χ</mi> <mo>→</mo> </mover> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> </mrow> </mrow> <annotation> $overrightarrow{chi }(D)=k$</annotation> </semantics></math> and each proper subdigraph <math> <semantics> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> <annotation> $H$</annotation> </semantics></math> of <math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> <annotation> $D$</annotation> </semantics></math> satisfies <mat

数图的二色数是指为数图顶点着色所需的最少颜色数，使得每个颜色类都能诱导出一个无环子数图。如果且的每个适当的子图均满足，则一个数图是无色的。对于整数和，我们定义（respect.科斯托奇卡和斯蒂比茨证明了。他们还猜想存在一个常数，对于和足够大。众所周知，这一猜想对于 .在本研究中，我们证明了每一个顶点上的 4-临界定向图都至少有弧，从而证明了对 ...的猜想。我们还精确地描述了顶点上具有精确弧的 4-dicritical 数字图的特征。

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引用次数: 0

Counting triangles in regular graphs 计算规则图形中的三角形

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-25 DOI: 10.1002/jgt.23156

Jialin He, Xinmin Hou, Jie Ma, Tianying Xie

In this paper, we investigate the minimum number of triangles, denoted by <math> <semantics> <mrow> <mrow> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $t(n,k)$</annotation> </semantics></math>, in <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math>-vertex <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> <annotation> $k$</annotation> </semantics></math>-regular graphs, where <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math> is an odd integer and <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> <annotation> $k$</annotation> </semantics></math> is an even integer. The well-known Andrásfai–Erdős–Sós Theorem has established that <math> <semantics> <mrow> <mrow> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>></mo> <mn>0</mn> </mrow> </mrow> <an

在本文中，我们将研究有顶点不规则图形中三角形的最小数量，用表示，其中为奇数整数，为偶数整数。著名的 Andrásfai-Erdős-Sós 定理证明，如果 .在一项引人注目的工作中，Lo 提供了足够大的 , 的精确值，即 .在这里，我们通过确定整个范围内的精确值，弥补了上述结果之间的差距。这证实了康比、德-乔尼斯-德-韦尔克洛斯和康对足够大的 .

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引用次数: 0