Journal of Graph Theory最新文献

On the Minimum Number of Inversions to Make a Digraph k -(Arc-)Strong 使有向图k -（弧-）强的最小反转数

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-12-09 DOI: 10.1002/jgt.23290

Julien Duron, Frédéric Havet, Florian Hörsch, Clément Rambaud

<div> The inversion of a set <math> <semantics> <mrow> <mi>X</mi> </mrow> </semantics></math> of vertices in a digraph <math> <semantics> <mrow> <mi>D</mi> </mrow> </semantics></math> consists of reversing the direction of all arcs of <math> <semantics> <mrow> <mi>D</mi> <mrow> <mo>〈</mo> <mi>X</mi> <mo>〉</mo> </mrow> </mrow> </semantics></math>. We study <math> <semantics> <mrow> <mtext>sin</mtext> <msubsup> <mi>v</mi> <mi>k</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (resp., <math> <semantics> <mrow> <mtext>sin</mtext> <msub> <mi>v</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>) which is (for some positive integer <math> <semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>) the minimum number of inversions needed to transform <math> <semantics> <mrow> <mi>D</mi> </mrow> </semantics></math> into a <math> <semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>-arc-strong (resp., <math> <semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>-strong) digraph or <math> <semantics> <mrow> <mo>+</mo> <mi>∞</mi> </mrow> </semantics></math> if no such transformation exists. Note that <math> <semantics> <mrow> <mtext>sin</mtext> <msubsup> <mi>v</mi> <mi>k</mi> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow>

有向图D中顶点集合X的反转包括反转D < X >的所有弧线的方向。我们学习sink ' (D),sin vk (D)它是（对于某个正整数k）对D进行变换所需的最小逆序个数变成了一个k -弧强（音）。， k -强)有向图或+∞，如果不存在这样的变换。注意，sin v k ' (D)≤sin v k (D)。设sinvk ‘ (n) = max {sinvk ’(D)∣D是一个阶n}的2k边连通有向图。我们展示了以下结果，其中k是(i)−(vi)的固定整数：1 . 1 2 log （n−k + 1）≤sinvk ' (n)≤log n +每n≥k为4 k−3；2。对于任意正整数t，决定一个给定的有向图D是否有sinv k ' (D) &lt； +∞满足sinvk ' (D))≤t为np完全；3。对于任意正整数t，决定一个给定的有向图D是否带有sinv k (D)+∞满足sinvk (D)≤t为np完全；iv.如果T是顺序至少为2k + 1的锦标赛，则sinvk (T)≤2k，sin v k ' (T)≤4K + 0 (K)；v。 1 2log (2k + 1)≤sinvk(T)对于某个2k阶的锦标赛T+ 1；vi.如果T是阶数至少为19k−2的比武赛。， 11 k−2)，则sinvk (T)≤1 (p。， sinvk (T)≤3)；7。对于每一个λ &gt； 0，存在C，使得对于每一个正整数k和每一个锦标赛T至少2k + 1 + k个顶点，我们有sinvk (T)≤C。

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引用次数: 0

An Approach to the Girth Problem in Cubic Graphs 三次图中周长问题的一种方法

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-10-29 DOI: 10.1002/jgt.23289

Aya Bernstine, Nati Linial

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math>-vertex cubic graphs is attained by a 2-connected graph <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> </semantics></math>. By Petersen's graph theorem, <math> <semantics> <mrow> <mrow> <mi>E</mi> </mrow> </mrow> </semantics></math> is the disjoint union of a 2-factor and a perfect matching <math> <semantics> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> </semantics></math>. We refer to the edges of <math> <semantics> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> </semantics></math> as ribs and classify the cycles in <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> by their number of ribs. We define <math> <semantics> <mrow> <mrow> <msub> <mi>γ</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow>

我们提供了一个新的，渐进的方法来解决三次图的最大周长问题。很容易观察到，所有n顶点三次图的最大可能周长是由2连通图G = (V, e)。根据Petersen图定理，E是2因子与完美匹配M的不相交并。我们将M的边称为棱，并根据棱的数量对G中的环进行分类。我们定义γ k (n)是最小的整数g使得每一个n顶点的三次图都有一个给定的完美匹配M它的循环长度最多为g，最多有k个肋。这里，当k = 1时，我们确定这个函数直到很小的附加常数，对于较大的k， 2和一个小的乘法常数。

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引用次数: 0

Wheel-Like Bricks and Minimal Matching Covered Graphs 轮状砖块和最小匹配覆盖图

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-09-30 DOI: 10.1002/jgt.23288

Xiaoling He, Fuliang Lu, Jinxin Xue

<div> A connected graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> with at least two vertices is matching covered if each of its edges lies in a perfect matching. We say that an edge <math> <semantics> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> </semantics></math> in a matching covered graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is removable if <math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>−</mo> <mi>e</mi> </mrow> </mrow> </semantics></math> is matching covered. A pair <math> <semantics> <mrow> <mrow> <mrow> <mo>{</mo> <mrow> <mi>e</mi> <mo>,</mo> <mi>f</mi> </mrow> <mo>}</mo> </mrow> </mrow> </mrow> </semantics></math> of edges of a matching covered graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is a removable doubleton if <math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>−</mo> <mi>e</mi> <mo>−</mo> <mi>f</mi> </mrow> </mrow> </semantics></math> is matching covered, but neither <math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>−</mo> <mi>e</mi> </mrow> </mrow> </semantics></math> nor <math> <semantics>

一个至少有两个顶点的连通图G，如果它的每条边都是完全匹配的，那么它就是匹配覆盖的。我们说匹配覆盖图G中的边e是可移动的，如果G−e是匹配覆盖。一对{e，匹配覆盖图G的f}条边是可移动双元，如果G−e−f是匹配的，但G−e和G−f都不是。可移动边和可移动双子被称为可移动类，由Lovász和Plummer在匹配覆盖图的耳分解中引入。一个3连通图是一块砖，如果去掉任意两个不同的顶点，左图就有一个完美的匹配。如果G有顶点h，那么砖G就是类轮的，使得G的每一个可移动类都有一条边与h相关。Lucchesi和Murty提出了一个描述轮式砖块的问题。我们证明了每一个类轮砖都可以通过下面的简单图是奇数轮的图按一定的方式拼接得到。如果移除任意一条边，则匹配覆盖图最小，左侧图不匹配覆盖。Lovász和Plummer在1977年通过耳分解证明了不同于k2的最小匹配覆盖二部图的最小度为2。利用轮状砖的性质，证明了除K 2以外的最小匹配覆盖图的最小度为2或3。

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引用次数: 0

Brooks-Type Colourings of Digraphs in Linear Time 线性时间有向图的brooks型着色

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-09-05 DOI: 10.1002/jgt.23266

Daniel Gonçalves, Lucas Picasarri-Arrieta, Amadeus Reinald

<div> Brooks' Theorem is a fundamental result on graph colouring, stating that the chromatic number of a graph is almost always upper bounded by its maximal degree. Lovász showed that such a colouring may then be computed in linear time when it exists. Many analogues are known for variants of (di)graph colouring, notably for list-colouring and partitions into subgraphs with prescribed degeneracy. One of the most general results of this kind is due to Borodin, Kostochka, and Toft, when asking for classes of colours to satisfy ‘variable degeneracy’ constraints. An extension of this result to digraphs has recently been proposed by Bang-Jensen, Schweser, and Stiebitz, by considering colourings as partitions into ‘variable weakly degenerate’ subdigraphs. Unlike earlier variants, there exists no linear-time algorithm to produce colourings for these generalisations. We introduce the notion of (variable) bidegeneracy for digraphs, capturing multiple (di)graph degeneracy variants. We define the corresponding concept of <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math>-dicolouring, where <math> <semantics> <mrow> <mrow> <mi>F</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> is a vector of functions, and an <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math>-dicolouring requires vertices coloured <math> <semantics> <mrow> <mrow> <mi>i</mi> </mrow> </mrow> </semantics></math> to induce a ‘strictly-<math> <s

布鲁克斯定理是关于图着色的一个基本结论，指出图的色数几乎总是以它的最大度为上界。Lovász表明，这样的着色可以在线性时间内计算出来。已知许多类似的（di）图着色的变体，特别是列表着色和划分为具有规定退化的子图。这类最普遍的结果之一是由Borodin， Kostochka和Toft提出的，他们要求颜色的类别满足“可变简并性”约束。最近Bang-Jensen、Schweser和Stiebitz提出了将这一结果扩展到有向图的方法，他们将着色视为“可变弱简并”子有向图的分区。与早期的变体不同，不存在线性时间算法来为这些泛化生成着色。我们引入有向图的（变量）双降的概念，捕获多个（di）图退化变体。我们定义了相应的F -变色概念，其中F = （F 1）， f (s)是一个函数向量，而F -变色要求顶点i被着色，以诱导出一个严格- F i-bidegenerate subdigraph。我们证明了布鲁克斯的F -变色定理的一个类似物，推广了Bang-Jensen等人的结果，以及之前的类似物。我们的新方法提供了一个线性时间算法，给定一个有向图D，要么产生D的F变色，要么正确地证明不存在。这产生了第一个线性时间算法来计算（di）着色，对应于前面提到的布鲁克斯定理的推广。反过来，它给出了一个统一的框架来计算布鲁克斯定理的各种中间推广，如列表-(di)着色和划分为（可变）退化子（di）图。

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引用次数: 0

Spectral Extrema of Graphs With Fixed Size: Forbidden a Fan Graph, a Friendship Graph, or a Theta Graph 固定大小图的谱极值：禁止风扇图、友谊图或Theta图

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-08-28 DOI: 10.1002/jgt.23287

Shuchao Li, Sishu Zhao, Lantao Zou

<div> It is well-known that Brualdi-Hoffman-Turán-type problem inquiries about the maximum spectral radius <math> <semantics> <mrow> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> of an <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math>-free graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> with <math> <semantics> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> </semantics></math> edges. This can be regarded as a spectral characterization of the existence of the subgraph <math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math> within <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>. A significant contribution to this problem was made by Nikiforov (2002). He proved that <math> <semantics> <mrow> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>⩽</mo> <msqrt> <mrow> <mn>2</mn> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>∕</mo>

众所周知，Brualdi-Hoffman-Turán-type问题询问的是an的最大谱半径λ (G)有m条边的无F图G。这可以看作是子图F在G内存在的谱表征。Nikiforov（2002）对这个问题做出了重大贡献。他证明了λ (G)≤2m(1−1∕r)，每K r + 1大小为m的自由图。设θ 1 p，Q是图，它是通过连接两个顶点得到的，这两个顶点有三条内部不相交的路径，长度为1 p，分别是Q。设F k为扇形图，即，k1和路径kp−的连接1，令F k，3是k个三角形共用一个顶点得到的友谊图。本文利用k核方法和谱技术来解决固定大小图的谱极值问题。其次，我们表明，对于m或9 4 k 6 + 6k5 + 46 k4 + 56k3 + 196k2， K大于或等于3，如果G是F k，3个尺寸为m的自由；λ (G)≥k−1 + 4 m−k2 + 1 2。当且仅当G≠K K∨(M k−k−1(1) k；这证实了Li等人提出的一个猜想。最后，我们确定了θ 1 p，大小为m的无Q图具有最大的谱半径，q或p或3和p + q在哪里大于或等于2k + 1。并提出了进一步研究的问题。

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引用次数: 0

Maximal Spectral Radius of Minimally k -(Edge)-Connected Graphs 最小k（边）连通图的最大谱半径

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-07-31 DOI: 10.1002/jgt.23286

Mingqing Zhai, Huiqiu Lin, Jinlong Shu

<div> Minimally <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>-connected graphs are the main focus of both structural and extremal graph theory. Perhaps the most heavily investigated parameter of this graph family is the number <math> <semantics> <mrow> <mrow> <mo>∣</mo> <msub> <mi>V</mi> <mi>k</mi> </msub> <mo>∣</mo> </mrow> </mrow> </semantics></math> of vertices of degree <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>. Mader proved a tight lower bound for <math> <semantics> <mrow> <mrow> <mo>∣</mo> <msub> <mi>V</mi> <mi>k</mi> </msub> <mo>∣</mo> </mrow> </mrow> </semantics></math>, independent of <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>, and the order <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math>. In 1981, inspired by matroids, Oxley discovered that in many cases, a considerably better bound can be given by using the size <math> <semantics> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> </semantics></math> as a parameter. Along this line, Schmidt [Tight bounds for the vertices of degree <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math> in minimally <math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>-connected graphs, J.

最小k连通图是结构图论和极值图论的主要研究方向。也许这个图族中研究得最多的参数是次数顶点的个数∣vk∣k .Mader证明了一个与k无关的严密下界，(n)1981年，受拟阵的启发，奥克斯利发现，在许多情况下，使用大小m作为参数可以给出一个更好的边界。沿着这条线，Schmidt[最小k连通图中k次顶点的紧界，J.图论88(2018)146-153]证明了∣V k∣≥max {（k + 1） n−2m；(m−n+ k)∕(K−1)²}，这个边界是最好的。另一个有趣的问题是对于固定大小的连通图：在m条边上的最小k（边）连通图的最大谱半径是多少？这一贡献可以追溯到Brualdi和Hoffman，他还推测gm是所有有m条边的连通图中的极值图，其中G m是由完全图K s by得到的添加一个度为m - s2的新顶点。1988年，罗林森用双特征向量变换完全解决了这个猜想。最近，Lou， Gao和Huang（2023）回答了k = 2的情况。本文彻底解决了这一问题，并进一步对于k≥2且m = Ω (K 6)时，确定了唯一极值图。

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引用次数: 0

Tight Bounds for Rainbow Partial F -Tiling in Edge-Colored Complete Hypergraphs 边色完全超图中彩虹部分F -平铺的紧界

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-07-23 DOI: 10.1002/jgt.23282

Jinghua Deng, Jianfeng Hou, Xizhi Liu, Caihong Yang

For an $� � � � � r � � �$ -graph $� � � � � F � � �$ and integers $� � � � � n � �, � � t � � �$ satisfying $� � � � � t � � \leq � � n � � / � � v � � � (� � F � �) � � � �$ , let $� � � � � ar � � � (� � � n � �, � � t � � F � � �) � � � �$ denote the minimum integer $� � � � � N � � �$ such that every edge-coloring of $� � � � � �_{K}^{� �} � � �$ using $� �$

对于r -图F和整数n，t满足t≤n / v (F)，让ar (n)t (F)表示最小整数N，使得的每条边着色K n r使用n种颜色包含t的彩虹副本F ,t F是由t个顶点不相交的拷贝组成的r个图F .当t = 1时，是Erdős-Simonovits-Sós[1]提出的经典反拉姆齐问题。当F为单条边时，这就成为Schiermeyer[2]和Özkahya-Young[3]引入的彩虹匹配问题。我们对ar (n)进行了系统研究，t (F)对于t远小于例(n)；F) / n r−1。我们的第一个主要结果提供了ar (n)的减少，从F到n， 2 F)，当F是有界光滑的，大多数先前研究过的超图都满足这两个性质。作为第一个结果的补充，第二个主要结果利用Turán数字之间的间隔确定了ar (n)，t F)相对较小的t。这两个结果一起决定了ar (n)t F)对于一大类超图。此外，后一种结果具有适用于未知Turán密度的超图的优点，例如著名的四面体k43。

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引用次数: 0

Dominating K t -Models 支配K - t模型

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-07-22 DOI: 10.1002/jgt.23272

Freddie Illingworth, David R. Wood

A dominating <math> <semantics> <mrow> <mrow> <msub> <mi>K</mi> <mi>t</mi> </msub> </mrow> </mrow> </semantics></math>-model in a graph <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is a sequence <math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>T</mi> <mi>t</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> of pairwise disjoint non-empty connected subgraphs of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>, such that for <math> <semantics> <mrow> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>⩽</mo> <mi>t</mi> </mrow> </mrow> </semantics></math> every vertex in <math> <semantics> <mrow> <mrow> <msub> <mi>T</mi> <mi>j</m

图G中的主导K - t模型是一个序列(1)……的成对不相交非空连通子图的T (TG ,使得对于1≤I &lt； j≤t中的每个顶点tj在t1中有一个邻居。将tj中的每个顶点替换为tj中的某个顶点检索K - t模型的标准定义，这就相当于K t是G的次元。我们探索在什么意义上支配K - t -模型表现得像（非支配）Kt - 模型。这两个概念对于t≤3是等价的，但是对于t =已经非常不同了4，因为任何图的1-细分都没有支配的k4 -模型。然而，我们证明了每一个没有主导k4模型的图都是2-简并的和3-可着色的。更普遍的是,我们证明了每一个没有主导K - t模型的图都是2T−2可着色。基于与色数的联系，我们研究了无主导K - t模型的图的最大平均度。我们给出了2t - 2的上界并证明了随机图给出(1−0(1)的下界t log t，我们推测它是渐近紧的。这个结果与K - t -minor-free的情况相反，最大平均度是Θ (t log t) . 哈德维格的猜想自然得到了加强：是否每一个没有支配K的图(T−1)-可着色？我们为此提供了两点证据：(1)几乎对每个图都是正确的。(2)每一个没有支配K t模型的图G都有a（t−1）-可着色诱导子图一半的顶点，这意味着存在一个独立的集合，其大小至少为b| V (G) | 2 t−2。

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引用次数: 0

A Strengthening on Consecutive Odd Cycles in Graphs of Given Minimum Degree 最小次给定图中连续奇环的一个强化

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-07-21 DOI: 10.1002/jgt.23281

Hao Lin, Guanghui Wang, Wenling Zhou

Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every 2-connected non-bipartite graph with minimum degree at least

� � �                  � � k �                     � + �                     � 1 � � �

contains

� � �                  � � � ⌈ �                        � � k �                           � / �                           � 2 � �                        � ⌉ � � � �

cycles with consecutive odd lengths. In particular, they showed that this conjecture holds when

� � �                  � � k � � �

is even. In this paper, we confirm this conjecture for any

� � �                  � � k �                     � \in �                     � N � � �

. Moreover, we also improve some previous results about cycles of consecutive lengths.

[J]。Combin。Ser的理论。B,[2018]推测每一个最小度数至少为k + 1的2连通非二部图都包含有K / 2个连续奇数长度的环。特别地，他们证明了当k是偶数时这个猜想成立。在本文中，我们对任意k∈N证实了这个猜想。此外，我们还改进了前人关于连续长度循环的一些结果。

{"title":"A Strengthening on Consecutive Odd Cycles in Graphs of Given Minimum Degree","authors":"Hao Lin, Guanghui Wang, Wenling Zhou","doi":"10.1002/jgt.23281","DOIUrl":"https://doi.org/10.1002/jgt.23281","url":null,"abstract":"<div>\u0000 \u0000 Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every 2-connected non-bipartite graph with minimum degree at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> cycles with consecutive odd lengths. In particular, they showed that this conjecture holds when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is even. In this paper, we confirm this conjecture for any <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>N</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Moreover, we also improve some previous results about cycles of consecutive lengths.\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 4","pages":"431-436"},"PeriodicalIF":1.0,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145272670","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

On Min-Bisections of Graphs 关于图的最小等分

IF 1 3区数学 Q2 MATHEMATICS

Journal of Graph Theory

Pub Date : 2025-07-21 DOI: 10.1002/jgt.23284

Jianfeng Hou, Shufei Wu

<div> A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. Let <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> be a graph with <math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math> vertices and <math> <semantics> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> </semantics></math> edges. Bollobás and Scott asked the following: What are the largest and smallest cuts that we can guarantee with bisections of <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>? There are reasonable sufficient conditions such that <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> has bisections of size at least <math> <semantics> <mrow> <mrow> <mi>m</mi> <mo>/</mo> <mn>2</mn> <mo>+</mo> <mi>c</mi> <mi>n</mi> </mrow> </mrow> </semantics></math> for some <math> <semantics> <mrow> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </mrow> </semantics></math>. In this paper, we study the Min-Bisection problem which has arisen in numerous contexts, and initially give some sufficient conditions such that <math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> has bisections of size at most <math> <semantics> <mrow> <mrow> <mi>m</mi>

图的等分是其顶点集的二分，其中两个部分的顶点数最多相差1，其大小是穿过这两个部分的边的数量。设G是一个有n个顶点和m条边的图。Bollobás和Scott问了下面的问题：用G的等分我们能保证的最大和最小的切点是什么？有合理的充分条件使G的等分大小至少为m / 2 + cN对于某个c >； 0。在本文中，我们研究了在许多情况下出现的最小对分问题，并初步给出G的等分大小不超过m / 2−c的充分条件N对于某个c >； 0。

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引用次数: 0

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