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A State Sum for the Total Face Color Polynomial
人脸总颜色多项式的状态和
IF 0.9
3区 数学
Scott Baldridge, Louis H. Kauffman, Ben McCarty
{"title":"A State Sum for the Total Face Color Polynomial","authors":"Scott Baldridge, Louis H. Kauffman, Ben McCarty","doi":"10.1002/jgt.23239","DOIUrl":"https://doi.org/10.1002/jgt.23239","url":null,"abstract":"<div>\u0000 \u0000 <p>The total face color polynomial is based upon the Poincaré polynomials of a family of filtered <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-color homologies. It counts the number of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-face colorings of ribbon graphs for each positive integer <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. As such, it may be seen as a successor of the Penrose polynomial, which at <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> counts 3-edge colorings (and consequently 4-face colorings) of planar trivalent graphs. In this paper, we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"481-491"},"PeriodicalIF":0.9,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256576","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
Edge-Connectivity Between Edge-Ends of Infinite Graphs
无穷图的边端间的边连通性
IF 0.9
3区 数学
Leandro Aurichi, Lucas Real
{"title":"Edge-Connectivity Between Edge-Ends of Infinite Graphs","authors":"Leandro Aurichi, Lucas Real","doi":"10.1002/jgt.23234","DOIUrl":"https://doi.org/10.1002/jgt.23234","url":null,"abstract":"<p>In infinite graph theory, the notion of <i>ends</i>, first introduced by Freudenthal and Jung for locally finite graphs, plays an important role when generalizing statements from finite graphs to infinite ones. Nash-Williams' Tree-Packing Theorem and MacLane's Planarity Criteria are examples of results that allow a topological approach, in which ends may be considered as <i>endpoints</i> of rays. In fact, there are extensive studies in the literature showing that classical (vertex-)connectivity theorems for finite graphs can be discussed regarding ends, in a more general context. However, aiming to generalize results of edge-connectivity, this paper recalls the definition of <i>edge-ends</i> in infinite graphs due to Hahn, Laviolette and Širáň. In terms of that object, we state an edge version of Menger's Theorem (following a previous work of Polat) and generalize the Lovász-Cherkassky Theorem for infinite graphs with edge-ends (inspired by a recent paper of Jacobs, Joó, Knappe, Kurkofka and Melcher).</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"454-465"},"PeriodicalIF":0.9,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23234","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256574","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 Three Conjectures of Thomassen and the Extremal Digraphs for Two Conjectures of Nash-Williams
论托马森的三个猜想和纳什-威廉姆斯两个猜想的极值有向图
IF 0.9
3区 数学
Samvel Kh. Darbinyan
{"title":"On Three Conjectures of Thomassen and the Extremal Digraphs for Two Conjectures of Nash-Williams","authors":"Samvel Kh. Darbinyan","doi":"10.1002/jgt.23233","DOIUrl":"https://doi.org/10.1002/jgt.23233","url":null,"abstract":"<div>\u0000 \u0000 <p>Thomassen suggested the following three conjectures: (1) Every 2-strong <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-regular digraph of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, except for two exceptional digraphs of orders 5 and 7, is Hamiltonian. (2) Every 3-strong digraph of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and with a minimum degree of at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is Hamiltonian-connected. (3) Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a 4-strong digraph of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> such that the sum of the degrees of every pair of nonadjacent vertices is at least <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Then <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is Hamiltonian-connected. In this paper, we disprove Conjectures 1 and 2. We prove that: Conjecture 3 is true if and only if every 3-strong dig","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"412-425"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256533","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
The Generic Circular Triangle-Free Graph
一般圆无三角形图
IF 0.9
3区 数学
Manuel Bodirsky, Santiago Guzmán-Pro
{"title":"The Generic Circular Triangle-Free Graph","authors":"Manuel Bodirsky, Santiago Guzmán-Pro","doi":"10.1002/jgt.23235","DOIUrl":"https://doi.org/10.1002/jgt.23235","url":null,"abstract":"<p>In this article, we introduce the generic circular triangle-free graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and propose a finite axiomatization of its first-order theory. In particular, our main results show that a countable graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> embeds into <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> if and only if it is a <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>3</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <msub>\u0000 <mi>C</mi>\u0000 \u0000 <mn>5</mn>\u0000 </msub>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"426-445"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23235","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256303","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
Note on Hamiltonicity of Basis Graphs of Even Delta-Matroids
关于偶阵基图的哈密性的注记
IF 0.9
3区 数学
Donggyu Kim, Sang-il Oum
{"title":"Note on Hamiltonicity of Basis Graphs of Even Delta-Matroids","authors":"Donggyu Kim, Sang-il Oum","doi":"10.1002/jgt.23237","DOIUrl":"https://doi.org/10.1002/jgt.23237","url":null,"abstract":"<p>We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> sharing a common end, it has a Hamiltonian cycle using <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and avoiding <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, and each edge belongs to cycles of every length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"446-453"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23237","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256304","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
A Sharper Ramsey Theorem for Constrained Drawings
约束图的一个sharperramsey定理
IF 0.9
3区 数学
Pavel Paták
{"title":"A Sharper Ramsey Theorem for Constrained Drawings","authors":"Pavel Paták","doi":"10.1002/jgt.23226","DOIUrl":"https://doi.org/10.1002/jgt.23226","url":null,"abstract":"<p>Given a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and a collection <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of subsets of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mi>d</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> indexed by the subsets of vertices of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, a constrained drawing of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a drawing where each edge is drawn inside some set from <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, in such a way that nonadjacent edges are drawn in sets with disjoint indices. In this paper we prove a Ramsey-type result for such drawings. Furthermore, we show how the results can be used to obtain Helly-type theorems. More precisely, we prove the following. For each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, there is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>N</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>O</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msup>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"401-411"},"PeriodicalIF":0.9,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23226","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256493","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
Berge's Conjecture for Cubic Graphs With Small Colouring Defect
具有小着色缺陷的三次图的Berge猜想
IF 0.9
3区 数学
Ján Karabáš, Edita Máčajová, Roman Nedela, Martin Škoviera
{"title":"Berge's Conjecture for Cubic Graphs With Small Colouring Defect","authors":"Ján Karabáš, Edita Máčajová, Roman Nedela, Martin Škoviera","doi":"10.1002/jgt.23231","DOIUrl":"https://doi.org/10.1002/jgt.23231","url":null,"abstract":"<div>\u0000 \u0000 <p>A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for 3-edge-colourable cubic graphs, but remains widely open for graphs that are not 3-edge-colourable. The aim of this paper is to verify the validity of Berge's conjecture for cubic graphs that are in a certain sense close to 3-edge-colourable graphs. We measure the closeness by looking at the colouring defect, which is defined as the minimum number of edges left uncovered by any collection of three perfect matchings. While 3-edge-colourable graphs have defect 0, every bridgeless cubic graph with no 3-edge-colouring has defect at least 3. In 2015, Steffen proved that the Berge conjecture holds for cyclically 4-edge-connected cubic graphs with colouring defect 3 or 4. Our aim is to improve Steffen's result in two ways. We show that all bridgeless cubic graphs with defect 3 satisfy Berge's conjecture irrespectively of their cyclic connectivity. If, additionally, the graph in question is cyclically 4-edge-connected, then four perfect matchings suffice, unless the graph is the Petersen graph. The result is best possible as there exists an infinite family of cubic graphs with cyclic connectivity 3 which have defect 3 but cannot be covered with four perfect matchings.</p></div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"387-396"},"PeriodicalIF":0.9,"publicationDate":"2025-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944819","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
Conformally Rigid Graphs
共形刚性图
IF 0.9
3区 数学
Stefan Steinerberger, Rekha R. Thomas
{"title":"Conformally Rigid Graphs","authors":"Stefan Steinerberger, Rekha R. Thomas","doi":"10.1002/jgt.23229","DOIUrl":"https://doi.org/10.1002/jgt.23229","url":null,"abstract":"<div>\u0000 \u0000 <p>Given a finite, simple, connected 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> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, we consider the associated graph Laplacian matrix <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>A</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with eigenvalues <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <msub>\u0000 <mi>λ</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo><</mo>\u0000 \u0000 <msub>\u0000 <mi>λ</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mo>⋯</mo>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <msub>\u0000 <mi>λ</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. One can also consider the same graph equipped with positive edge weights <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>w</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"366-386"},"PeriodicalIF":0.9,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944680","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
Eigenvalue Approach to Dense Clusters in Hypergraphs
IF 0.9
3区 数学
Yuly Billig
{"title":"Eigenvalue Approach to Dense Clusters in Hypergraphs","authors":"Yuly Billig","doi":"10.1002/jgt.23218","DOIUrl":"https://doi.org/10.1002/jgt.23218","url":null,"abstract":"<p>In this article, we investigate the problem of finding in a given weighted hypergraph a subhypergraph with the maximum possible density. Using the notion of a support matrix we prove that the density of an optimal subhypergraph is equal to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∥</mo>\u0000 \u0000 <msup>\u0000 <mi>A</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msup>\u0000 \u0000 <mi>A</mi>\u0000 \u0000 <mo>∥</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> for an optimal support matrix <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Alternatively, the maximum density of a subhypergraph is equal to the solution of a minimax problem for column sums of support matrices. We study the density decomposition of a hypergraph and show that it is a significant refinement of the Dulmage–Mendelsohn decomposition. Our theoretical results yield an efficient algorithm for finding the maximum density subhypergraph and more generally, the density decomposition for a given weighted hypergraph.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"353-365"},"PeriodicalIF":0.9,"publicationDate":"2025-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23218","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944639","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
List Packing and Correspondence Packing of Planar Graphs
IF 0.9
3区 数学
Daniel W. Cranston, Evelyne Smith-Roberge
{"title":"List Packing and Correspondence Packing of Planar Graphs","authors":"Daniel W. Cranston, Evelyne Smith-Roberge","doi":"10.1002/jgt.23222","DOIUrl":"https://doi.org/10.1002/jgt.23222","url":null,"abstract":"<div>\u0000 \u0000 <p>For a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and a list assignment <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>L</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> for all <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-packing consists of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-colorings <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>φ</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>φ</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> such that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>φ</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"339-352"},"PeriodicalIF":0.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944528","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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