Journal of Graph Theory最新文献_第6页

Counting rainbow triangles in edge-colored graphs 计算边色图中的彩虹三角形

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-23 DOI: 10.1002/jgt.23158

Xueliang Li, Bo Ning, Yongtang Shi, Shenggui Zhang

Let <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> be an edge-colored graph on <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math> vertices. The minimum color degree of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>, denoted by <math> <semantics> <mrow> <mrow> <msup> <mi>δ</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${delta }^{c}(G)$</annotation> </semantics></math>, is defined as the minimum number of colors assigned to the edges incident to a vertex in <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>. In 2013, Li proved that an edge-colored graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> on <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math> vertices contains a rainbow triangle if <math> <semantics> <mrow> <mrow> <msup> <mi>δ</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </m

假设是一个顶点上的边色图。在 2013 年，Li 证明了如果......，则顶点上的边色图包含彩虹三角形。在本文中，我们获得了关于.NET 中通过一个给定顶点的彩虹三角形数量的几个估计值。因此，我们证明了边色图中彩虹三角形的计数结果。其中一个主要定理指出，彩虹三角形的数量至少为，考虑到彩虹部分图兰图的阶数可被。这意味着，如果，则有彩虹三角形；如果，则有彩虹三角形。这两个结果在大小阶的意义上都很严密。我们还证明了 Broersma 等人提出的彩色邻域联合条件下彩虹三角形的计数版定理，以及强迫彩色友谊子图（即共享一个共同顶点的彩虹三角形）的渐近紧密彩色度条件。

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引用次数: 0

Defective acyclic colorings of planar graphs 平面图形的缺陷非循环着色

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-21 DOI: 10.1002/jgt.23154

On-Hei Solomon Lo, Ben Seamone, Xuding Zhu

This paper studies two variants of defective acyclic coloring of planar graphs. For a graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> and a coloring <math> <semantics> <mrow> <mrow> <mi>φ</mi> </mrow> </mrow> <annotation> $varphi $</annotation> </semantics></math> of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>, a 2-colored cycle (2CC) transversal is a subset <math> <semantics> <mrow> <mrow> <msup> <mi>E</mi> <mo>′</mo> </msup> </mrow> </mrow> <annotation> ${E}^{^{prime} }$</annotation> </semantics></math> of <math> <semantics> <mrow> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $E(G)$</annotation> </semantics></math> that intersects every 2-colored cycle. Let <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> <annotation> $k$</annotation> </semantics></math> be a positive integer. We denote by <math> <semantics> <mrow> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${m}_{k}(G)$</annotation> </semantics></math> the minimum integer <math> <semantics> <mrow> <mrow>

本文研究平面图形缺陷非循环着色的两种变体。对于一个图和一个着色为 , 的双色循环 (2CC) 横向是与每个双色循环相交的子集。设为正整数。我们用最小整数来表示这样一个图，它有一个大小为的 2CC 横向，用最小大小来表示这样一个图的子集，它是非循环可着色的。我们证明了，对于任何-顶点 3-可着色的平面图，以及对于任何平面图，只要。我们证明了这些上界是尖锐的：有无限多的平面图可以达到这些上界。此外，最小 2CC 横向的选择方式可以诱导出一个森林。我们还证明，对于任何平面图且 .

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引用次数: 0

Cycles in 3-connected claw-free planar graphs and 4-connected planar graphs without 4-cycles 3连通无爪平面图和4连通无4周期平面图中的周期

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-21 DOI: 10.1002/jgt.23152

On-Hei Solomon Lo

The cycle spectrum <math> <semantics> <mrow> <mrow> <mi>CS</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${mathscr{CS}}(G)$</annotation> </semantics></math> of a graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> is the set of the cycle lengths in <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>. Let <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> ${mathscr{G}}$</annotation> </semantics></math> be a graph class. For any integer <math> <semantics> <mrow> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </mrow> <annotation> $kge 3$</annotation> </semantics></math>, define <math> <semantics> <mrow> <mrow> <msub> <mi>f</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${f}_{{mathscr{G}}}(k)$</annotation> </semantics></math> to be the least integer <math> <semantics> <mrow> <mrow> <msup> <mi>k</mi> <mo>′</mo> </msup> <mo>≥</mo> <mi>k</mi> </mrow> </mrow> <annotation> ${k}^{^{prime} }ge k$</annotation> </semantics>

图的循环谱是图中循环长度的集合。设为一个图类。对于任意整数 , 定义为最小整数，使得对于任意周长至少为 .用、和分别表示 3 连接平面图形类、3 连接立方平面图形类、3 连接无爪平面图形类和 3 连接无爪立方平面图形类。所有......的和值都是已知的。在本文的第一部分，我们通过给出和的值，证明了这些结果的无爪版本。在第二部分中，我们将研究无 4 循环的 4 连接平面图的循环谱。邦迪猜想每个 4 连接平面图都有从 3 到的所有循环长度，只有一个偶数长度除外。这一猜想的真实性意味着每个不含 4 循环的 4 连接平面图都具有 4 以外的所有循环长度。我们证明了可以嵌入平面图中，使得对于任意有一个循环且所有不在循环中的顶点都位于 .

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引用次数: 0

Local version of Vizing's theorem for multigraphs 多图维京定理的局部版本

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-16 DOI: 10.1002/jgt.23155

Clinton T. Conley, Jan Grebík, Oleg Pikhurko

Extending a result of Christiansen, we prove that every multigraph <math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>,</mo> <mi>E</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $G=(V,E)$</annotation> </semantics></math> admits a proper edge colouring <math> <semantics> <mrow> <mrow> <mi>ϕ</mi> <mo>:</mo> <mi>E</mi> <mo>→</mo> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mspace></mspace> </mrow> <mo>}</mo> </mrow> </mrow> </mrow> <annotation> $phi :Eto {1,2,ldots ,}$</annotation> </semantics></math> which is local, that is, <math> <semantics> <mrow> <mrow> <mi>ϕ</mi> <mrow> <mo>(</mo> <mi>e</mi> <mo>)</mo> </mrow> <mo>⩽</mo> <mi>max</mi> <mrow> <mo>{</mo> <mrow> <mi>d</mi> <mrow> <mo>(</mo>

通过扩展克里斯蒂安森的一个结果，我们证明了每一个多图都有一个适当的边着色，这个边着色是局部的，也就是说，对于每一条有端点的边，（resp. ）表示顶点的度数（resp. 最大边乘）。这是从范式方程的局部版本中推导出来的。

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引用次数: 0

Stability from graph symmetrization arguments in generalized Turán problems 广义图兰问题中图对称论证的稳定性

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-15 DOI: 10.1002/jgt.23151

Dániel Gerbner, Hilal Hama Karim

Given graphs <math> <semantics> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> <annotation> $H$</annotation> </semantics></math> and <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math>, <math> <semantics> <mrow> <mrow> <mtext>ex</mtext> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>H</mi> <mo>,</mo> <mi>F</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $text{ex}(n,H,F)$</annotation> </semantics></math> denotes the largest number of copies of <math> <semantics> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> <annotation> $H$</annotation> </semantics></math> in <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math>-free <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math>-vertex graphs. Let <math> <semantics> <mrow> <mrow> <mi>χ</mi> <mrow> <mo>(</mo> <mi>H</mi> <mo>)</mo> </mrow> <mo><</mo> <mi>χ</mi> <mrow> <mo>(</mo> <mi>F</mi> <mo>)</mo> </mrow

给定图和，表示无顶点图中的最大副本数。如果下面的条件成立，我们就说它是 F-Turán-stable 的。对于任意存在这样的图，如果一个无顶点图至少包含、的副本，那么和的编辑距离最多为。如果将图兰图替换为任何完整的-部分图后同样成立，我们就说图兰图具有弱F-图兰稳定性。众所周知，这种稳定性意味着几种情况下的精确结果。我们证明了色度数最多的完整多方图是弱-图兰稳定的。我们部分正面回答了莫里森、尼尔、诺林、拉扎耶夫斯基和韦索莱克的一个问题，证明了对于每个图，如果足够大，则是-图兰稳定的。最后，我们证明，如果是双向图，那么对于足够大的图，它是弱-图兰稳定的。

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引用次数: 0

Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles 多色图兰数 II：鲁兹萨-塞梅雷迪定理的一般化以及关于小群和奇数循环的新结果

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-14 DOI: 10.1002/jgt.23147

Benedek Kovács, Zoltán Lóránt Nagy

In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let <math> <semantics> <mrow> <mrow> <msub> <mtext>ex</mtext> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>G</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${text{ex}}_{F}(n,G)$</annotation> </semantics></math> denote the maximum number of edge-disjoint copies of a fixed simple graph <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math> that can be placed on an <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math>-vertex ground set without forming a subgraph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> whose edges are from different <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math>-copies. The case when both <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math> and <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when <math> <semantics> <mrow>

在本文中，我们将继续研究图兰的禁止子图问题和鲁兹萨-塞梅雷迪问题的自然概括。让表示一个固定简单图的边不相交副本的最大数目，这些副本可以放在一个-顶点地面集上，而不会形成一个边来自不同副本的子图。当和都是三角形时，基本上就可以得出鲁兹萨和塞梅雷迪的定理。我们应用厄尔多斯、弗兰克尔和罗德尔的一个数论结果，将他们的结果推广到和都是任意小块的情况。这一扩展反过来决定了一大系列图对的数量级，它们将是亚二次方的，但几乎是二次方的。由于要确定的线性均匀超图图兰问题构成了多色图兰问题的一个类别，根据同一性，我们的结果确定了每一个周长为 3 的图的线性超图图兰数，并且每一个图的线性超图图兰数都达到了亚对数因子。

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引用次数: 0

Thin edges in cubic braces 立方括号中的细边

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-14 DOI: 10.1002/jgt.23150

Xiaoling He, Fuliang Lu

For a vertex set <math> <semantics> <mrow> <mrow> <mi>X</mi> </mrow> </mrow> <annotation> $X$</annotation> </semantics></math> in a graph, the edge cut <math> <semantics> <mrow> <mrow> <mo>∂</mo> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $partial (X)$</annotation> </semantics></math> is the set of edges with exactly one end vertex in <math> <semantics> <mrow> <mrow> <mi>X</mi> </mrow> </mrow> <annotation> $X$</annotation> </semantics></math>. An edge cut <math> <semantics> <mrow> <mrow> <mo>∂</mo> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $partial (X)$</annotation> </semantics></math> is tight if every perfect matching of the graph contains exactly one edge in <math> <semantics> <mrow> <mrow> <mo>∂</mo> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $partial (X)$</annotation> </semantics></math>. A matching covered bipartite graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> is a brace if, for every tight cut <math> <semantics> <mrow> <mrow> <mo>∂</mo> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow>

对于图中的顶点集，边切是指在 ...中恰好有一个末端顶点的边的集合。如果图中的每个完美匹配都恰好包含一条边，则边切是紧密的。如果对于每个紧密切分 , 或 , 其中 , 一个匹配覆盖的二叉图是一个括号。在洛瓦兹对匹配覆盖图的紧切分解中，括号起着重要作用。在一个图中，度数为 2 的顶点恰好有两个相邻的和，对该顶点的双收缩包括将该顶点集收缩为单个顶点。匹配覆盖图的缩减图是重复二度顶点的二缩减后得到的图。如果的缩回是一个括号，那么至少有六个顶点的括号边就是细边。麦库艾格证明了每个至少有六个顶点的括号都有一条细边。卡瓦略、卢切西和穆尔蒂证明了阶数为六或更多的括号有两条细边，并猜想其中包含两条不相邻的细边。此外，他们还提出了一个更强的猜想：存在一个正常量，使得每一个长方体都有细边。通过证明在每一个阶数至少为六的立方括号中，存在一个大小至少为的匹配，使得其中的每一条边都是薄边，我们证明了上述两个猜想对于立方括号是成立的。

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引用次数: 0

Growth rates of the bipartite Erdős–Gyárfás function 双向厄尔多斯-京法斯函数的增长率

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-11 DOI: 10.1002/jgt.23149

Xihe Li, Hajo Broersma, Ligong Wang

Given two graphs <math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>,</mo> <mi>H</mi> </mrow> </mrow> <annotation> $G,H$</annotation> </semantics></math> and a positive integer <math> <semantics> <mrow> <mrow> <mi>q</mi> </mrow> </mrow> <annotation> $q$</annotation> </semantics></math>, an <math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>H</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $(H,q)$</annotation> </semantics></math>-coloring of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> is an edge-coloring of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> such that every copy of <math> <semantics> <mrow> <mrow> <mi>H</mi> </mrow> </mrow> <annotation> $H$</annotation> </semantics></math> in <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> receives at least <math> <semantics> <mrow> <mrow> <mi>q</mi> </mrow> </mrow> <annotation> $q$</annotation>

给定两个图和一个正整数 , 的-着色是一个边着色，使得 in 的每一个副本都得到至少不同的颜色。双向厄尔多斯-雅尔法函数的定义是，要有一个-着色所需的最少颜色数。在本文中，我们将研究该函数在不一定平衡的完整双方图中的渐近行为。我们的主要结果涉及该函数增长率的阈值、下限和上限，特别是（亚）线性增长和（亚）二次增长。我们还获得了平衡两方情况下的新下限，并改进了阿克森诺维奇、富雷迪和穆巴伊给出的几个结果。我们的证明技术是基于 Pohoata 和 Sheffer 最近开发的彩色能量法及其改进版，以及对 Corrádi 提出的一个旧结果的推广。

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引用次数: 0

Maximum odd induced subgraph of a graph concerning its chromatic number 关于图形色度数的最大奇数诱导子图

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-09 DOI: 10.1002/jgt.23148

Tao Wang, Baoyindureng Wu

Let <math> <semantics> <mrow> <mrow> <msub> <mi>f</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${f}_{o}(G)$</annotation> </semantics></math> be the maximum order of an odd induced subgraph of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math>. In 1992, Scott proposed a conjecture that <math> <semantics> <mrow> <mrow> <msub> <mi>f</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>≥</mo> <mfrac> <mi>n</mi> <mrow> <mi>χ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation> ${f}_{o}(G)ge frac{n}{chi (G)}$</annotation> </semantics></math> for a graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> of order <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math> without isolated vertices, where <math> <semantics> <mrow> <mrow> <mi>χ</mi>

让是 .的奇数诱导子图的最大阶数。 1992 年，斯科特（Scott）提出了一个猜想：对于无孤立顶点的阶数图，让是 .的色度数。在本文中，我们证明了这一猜想对于二叉图并不成立，但对于所有线图都成立。此外，我们还推翻了 Berman、Wang 和 Wargo 于 1997 年提出的猜想，即对于秩为 .斯科特的猜想对于色度数至少为 3 的图是开放的。

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引用次数: 0

Cliques in squares of graphs with maximum average degree less than 4 最大平均度数小于 4 的图形正方形中的小群

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2024-07-02 DOI: 10.1002/jgt.23125

Daniel W. Cranston, Gexin Yu

Hocquard, Kim, and Pierron constructed, for every even integer <math> <semantics> <mrow> <mrow> <mi>D</mi> <mo>≥</mo> <mn>2</mn> </mrow> </mrow> <annotation> $Dge 2$</annotation> </semantics></math>, a 2-degenerate graph <math> <semantics> <mrow> <mrow> <msub> <mi>G</mi> <mi>D</mi> </msub> </mrow> </mrow> <annotation> ${G}_{D}$</annotation> </semantics></math> with maximum degree <math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> <annotation> $D$</annotation> </semantics></math> such that <math> <semantics> <mrow> <mrow> <mi>ω</mi> <mrow> <mo>(</mo> <msubsup> <mi>G</mi> <mi>D</mi> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> <mi>D</mi> </mrow> </mrow> <annotation> $omega ({G}_{D}^{2})=frac{5}{2}D$</annotation> </semantics></math>. We prove for (a) all 2-degenerate graphs <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> and (b) all graphs <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation>

Hocquard、Kim 和 Pierron 为每一个偶整数，构造了一个具有最大度的 2 退化图，使得。我们证明了 (a) 所有 2-enerate图和 (b) 所有具有 , 的图的小群数的上界，这些小群数的上界与该构造给出的下界相匹配，且不超过小的加常数。我们证明，如果是最大度为 , 的 2畸变图，那么（当度足够大时），如果是最大度为 , 的 2畸变图，那么（当度足够大时）。而如果最大度为，那么。因此，霍夸尔等人的构造本质上是最好的。我们的证明引入了一种 "令牌传递 "技术，以推导出...中相邻顶点的非相邻性的关键信息。这是一种处理此类图的强大技术，以前从未在文献中出现过。

{"title":"Cliques in squares of graphs with maximum average degree less than 4","authors":"Daniel W. Cranston, Gexin Yu","doi":"10.1002/jgt.23125","DOIUrl":"10.1002/jgt.23125","url":null,"abstract":"Hocquard, Kim, and Pierron constructed, for every even integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $Dge 2$</annotation>\u0000 </semantics></math>, a 2-degenerate graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>D</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${G}_{D}$</annotation>\u0000 </semantics></math> with maximum degree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ω</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msubsup>\u0000 <mi>G</mi>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msubsup>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mfrac>\u0000 <mn>5</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 \u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $omega ({G}_{D}^{2})=frac{5}{2}D$</annotation>\u0000 </semantics></math>. We prove for (a) all 2-degenerate graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> and (b) all graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"559-577"},"PeriodicalIF":0.9,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23125","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531766","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