<p>The famous Dirac's theorem states that for each <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </mrow> </semantics></math> every <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math>-vertex graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> with minimum degree <span></span><math> <semantics> <mrow> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>≥</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> </mrow> </mrow> </semantics></math> has a Hamiltonian cycle. When <span></span><math> <semantics> <mrow> <mrow> <mi>δ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo><</mo> <mi>n</mi> <mo>/</mo> <mn>2</mn> </mrow> </mrow> </semantics></math>, this cannot be guaranteed, but the existence of some other specific subgraphs can be provided. Gargano, Hammar, Hell, Stacho, and Vaccaro proved that every connected <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math>-vertex graph <span></span><math> <semantics> <mrow>
著名的狄拉克定理指出，对于n≥3个n顶点的图最小度δ (G)≥n / 2有一个哈密顿循环。δ (G) < n /2，这不能保证，但可以提供一些其他特定子图的存在性。加尔加诺，哈马尔，地狱，斯塔乔，和Vaccaro证明了每一个带δ的连通n顶点图G(G)≥(n−1) / 3包含一个生成蜘蛛，即一棵最多有一个顶点度至少为3的生成树。 后来，陈、费拉拉、胡、雅各布森、Liu证明了一个更强（和准确）的结果，即对于n≥56，每个连通的n顶点图G与δ (G)≥(n−2)/ 3包含一个跨越扫帚，即通过将星形中心连接到路径的端点而获得的跨越蜘蛛。他们还证明了一个带δ (G)的二连通图G≥(n−2)/3和一些额外的属性包含一个生成水母，它是通过将恒星的中心粘合到与该恒星不相交的循环中的顶点而获得的图。请注意，每个跨越水母都包含一个跨越扫帚。 本文的目标是证明一个保证跨越水母存在的精确的oretype界：我们证明了如果G是一个有n个顶点的2连通图，使得每个不相邻的顶点对[au:]V满足d (u+ d (v)≥2 n−3 3，那么G有一个跨越水母。作为推论，我们得到了Chen等人的两个结果的强化：保证跨越水母存在的最小度条件和跨越扫帚存在的ore型充分条件。对于无穷多个n，推论是清晰的。我们证明的主要成分之一是对由伍德尔引起的跳跃引理的修正。
{"title":"Ore-Type Conditions for Existence of a Jellyfish in a Graph","authors":"Jaehoon Kim, Alexandr Kostochka, Ruth Luo","doi":"10.1002/jgt.70002","DOIUrl":"https://doi.org/10.1002/jgt.70002","url":null,"abstract":"<p>The famous Dirac's theorem states that for each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> every <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-vertex graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with minimum degree <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>δ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> has a Hamiltonian cycle. When <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>δ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>/</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, this cannot be guaranteed, but the existence of some other specific subgraphs can be provided. Gargano, Hammar, Hell, Stacho, and Vaccaro proved that every connected <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-vertex graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"111 4","pages":"124-143"},"PeriodicalIF":1.0,"publicationDate":"2026-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146256572","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}
Manoj Belavadi, Kathie Cameron, Ni Luh Dewi Sintiari
<p>The reconfiguration graph of the <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>-colorings of a graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>, denoted <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>R</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math>, is the graph whose vertices are the <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math>-colorings of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> and two colorings are adjacent in <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>R</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> if they differ in color on exactly one vertex. A graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is said to be recolorable if <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>R</mi> <mi>ℓ</mi>
图G的k次着色的重构图，表示R k (G)，顶点是G的k种颜色且两种颜色相邻的图是什么R k (G)如果它们在一个顶点上颜色不同。图G是可复色的，如果对于所有的r≥χ， G)是连通的(g) + 1。我们演示了如何使用图的模分解来证明它是可再着色的。特别地，我们证明了每一个（p5，菱形）无图，每（p5，房子，公牛）无图，每一次（P 5），c5，无共叉图是可重新着色的。如果一个图不能被模分解分解成单个顶点，那么它就是素数图。 对于素数图H，研究了判定H是否可色的复杂性和判定两个给定之间是否存在一条路径的复杂性k - R中的着色k (H)。假设G是图的一个遗传类。我们证明了如果G中的每个素数图的每个放大都是可重新着色的，那么G中的每个图都是可重新着色的。
{"title":"Recoloring via Modular Decomposition","authors":"Manoj Belavadi, Kathie Cameron, Ni Luh Dewi Sintiari","doi":"10.1002/jgt.70005","DOIUrl":"https://doi.org/10.1002/jgt.70005","url":null,"abstract":"<p>The reconfiguration graph of the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-colorings of a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, denoted <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mi>k</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 </semantics></math>, is the graph whose vertices are the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-colorings of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and two colorings are adjacent in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mi>k</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 </semantics></math> if they differ in color on exactly one vertex. A graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is said to be recolorable if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mi>ℓ</mi>\u0000","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"111 4","pages":"113-123"},"PeriodicalIF":1.0,"publicationDate":"2026-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.70005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146217231","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}
<div> <p>The <i>inversion</i> of a set <span></span><math> <semantics> <mrow> <mi>X</mi> </mrow> </semantics></math> of vertices in a digraph <span></span><math> <semantics> <mrow> <mi>D</mi> </mrow> </semantics></math> consists of reversing the direction of all arcs of <span></span><math> <semantics> <mrow> <mi>D</mi> <mrow> <mo>〈</mo> <mi>X</mi> <mo>〉</mo> </mrow> </mrow> </semantics></math>. We study <span></span><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., <span></span><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 <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>) the minimum number of inversions needed to transform <span></span><math> <semantics> <mrow> <mi>D</mi> </mrow> </semantics></math> into a <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>-arc-strong (resp., <span></span><math> <semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>-strong) digraph or <span></span><math> <semantics> <mrow> <mo>+</mo> <mi>∞</mi> </mrow> </semantics></math> if no such transformation exists. Note that <span></span><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) <； +∞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 >； 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。
{"title":"On the Minimum Number of Inversions to Make a Digraph \u0000 \u0000 \u0000 k\u0000 \u0000 -(Arc-)Strong","authors":"Julien Duron, Frédéric Havet, Florian Hörsch, Clément Rambaud","doi":"10.1002/jgt.23290","DOIUrl":"https://doi.org/10.1002/jgt.23290","url":null,"abstract":"<div>\u0000 \u0000 <p>The <i>inversion</i> of a set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 </semantics></math> of vertices in a digraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </semantics></math> consists of reversing the direction of all arcs of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mrow>\u0000 <mo>〈</mo>\u0000 <mi>X</mi>\u0000 <mo>〉</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. We study <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>sin</mtext>\u0000 <msubsup>\u0000 <mi>v</mi>\u0000 <mi>k</mi>\u0000 <mo>′</mo>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> (resp., <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>sin</mtext>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>) which is (for some positive integer <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </semantics></math>) the minimum number of inversions needed to transform <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </semantics></math> into a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </semantics></math>-arc-strong (resp., <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </semantics></math>-strong) digraph or <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>+</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </semantics></math> if no such transformation exists. Note that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>sin</mtext>\u0000 <msubsup>\u0000 <mi>v</mi>\u0000 <mi>k</mi>\u0000 <mo>′</mo>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"111 2","pages":"31-62"},"PeriodicalIF":1.0,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145772314","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}
<p>We offer a new, gradual approach to the <i>largest girth problem for cubic graphs</i>. It is easily observed that the largest possible girth of all <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math>-vertex cubic graphs is attained by a <i>2-connected</i> graph <span></span><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, <span></span><math> <semantics> <mrow> <mrow> <mi>E</mi> </mrow> </mrow> </semantics></math> is the disjoint union of a 2-factor and a perfect matching <span></span><math> <semantics> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> </semantics></math>. We refer to the edges of <span></span><math> <semantics> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> </semantics></math> as <i>ribs</i> and classify the cycles in <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> by their number of ribs. We define <span></span><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。
{"title":"An Approach to the Girth Problem in Cubic Graphs","authors":"Aya Bernstine, Nati Linial","doi":"10.1002/jgt.23289","DOIUrl":"https://doi.org/10.1002/jgt.23289","url":null,"abstract":"<p>We offer a new, gradual approach to the <i>largest girth problem for cubic graphs</i>. It is easily observed that the largest possible girth of all <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-vertex cubic graphs is attained by a <i>2-connected</i> graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</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>E</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. By Petersen's graph theorem, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>E</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is the disjoint union of a 2-factor and a perfect matching <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. We refer to the edges of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> as <i>ribs</i> and classify the cycles in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> by their number of ribs. We define <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>γ</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"111 1","pages":"17-26"},"PeriodicalIF":1.0,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23289","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145533559","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}
<div> <p>A connected graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> with at least two vertices is <i>matching covered</i> if each of its edges lies in a perfect matching. We say that an edge <span></span><math> <semantics> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> </semantics></math> in a matching covered graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is <i>removable</i> if <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>−</mo> <mi>e</mi> </mrow> </mrow> </semantics></math> is matching covered. A pair <span></span><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 <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is a <i>removable doubleton</i> if <span></span><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 <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>−</mo> <mi>e</mi> </mrow> </mrow> </semantics></math> nor <span></span><math> <semantics>
Daniel Gonçalves, Lucas Picasarri-Arrieta, Amadeus Reinald
<div> <p>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 <i>(variable) bidegeneracy</i> for digraphs, capturing multiple (di)graph degeneracy variants. We define the corresponding concept of <span></span><math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math>-dicolouring, where <span></span><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 <span></span><math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math>-dicolouring requires vertices coloured <span></span><math> <semantics> <mrow> <mrow> <mi>i</mi> </mrow> </mrow> </semantics></math> to induce a ‘strictly-<span></span><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。
{"title":"Brooks-Type Colourings of Digraphs in Linear Time","authors":"Daniel Gonçalves, Lucas Picasarri-Arrieta, Amadeus Reinald","doi":"10.1002/jgt.23266","DOIUrl":"https://doi.org/10.1002/jgt.23266","url":null,"abstract":"<div>\u0000 \u0000 <p>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 <i>(variable) bidegeneracy</i> for digraphs, capturing multiple (di)graph degeneracy variants. We define the corresponding concept of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-dicolouring, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mo>…</mo>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>s</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a vector of functions, and an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-dicolouring requires vertices coloured <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> to induce a ‘strictly-<span></span><math>\u0000 <s","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 4","pages":"496-513"},"PeriodicalIF":1.0,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145271724","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}
<div> <p>It is well-known that Brualdi-Hoffman-Turán-type problem inquiries about the maximum spectral radius <span></span><math> <semantics> <mrow> <mrow> <mi>λ</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> of an <span></span><math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math>-free graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> with <span></span><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 <span></span><math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> </semantics></math> within <span></span><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 <span></span><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>
{"title":"Spectral Extrema of Graphs With Fixed Size: Forbidden a Fan Graph, a Friendship Graph, or a Theta Graph","authors":"Shuchao Li, Sishu Zhao, Lantao Zou","doi":"10.1002/jgt.23287","DOIUrl":"https://doi.org/10.1002/jgt.23287","url":null,"abstract":"<div>\u0000 \u0000 <p>It is well-known that Brualdi-Hoffman-Turán-type problem inquiries about the maximum spectral radius <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>λ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-free graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> edges. This can be regarded as a spectral characterization of the existence of the subgraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> within <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. A significant contribution to this problem was made by Nikiforov (2002). He proved that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>λ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>⩽</mo>\u0000 \u0000 <msqrt>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>∕</mo>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 4","pages":"483-495"},"PeriodicalIF":1.0,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145272923","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}
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