Journal of Graph Theory最新文献

Berge's Conjecture for Cubic Graphs With Small Colouring Defect

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-03-16 DOI: 10.1002/jgt.23231

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 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.</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

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-03-11 DOI: 10.1002/jgt.23229

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 Given a finite, simple, connected graph <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 <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 <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 <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 <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

Journal of Graph Theory Pub Date : 2025-03-02 DOI: 10.1002/jgt.23218

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":"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 <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 <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.","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

Journal of Graph Theory Pub Date : 2025-02-24 DOI: 10.1002/jgt.23222

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 For a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and a list assignment <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with <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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, an <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-packing consists of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-colorings <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 <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}

引用次数: 0

Distribution of Vertices Required a High-Degree Condition on Partitions of Graphs Under Degree Constraints 度约束下图分区上顶点分布的高度条件

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-02-16 DOI: 10.1002/jgt.23228

Michitaka Furuya, Shun-ichi Maezawa

{"title":"Distribution of Vertices Required a High-Degree Condition on Partitions of Graphs Under Degree Constraints","authors":"Michitaka Furuya, Shun-ichi Maezawa","doi":"10.1002/jgt.23228","DOIUrl":"https://doi.org/10.1002/jgt.23228","url":null,"abstract":"<div>\u0000 \u0000 Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be a graph, and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>→</mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mo>…</mo>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be functions. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mn>1</mn>\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> be the union of edges shared by two cycles of order at most four, and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 \u0000 <mn>0</mn>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"315-331"},"PeriodicalIF":0.9,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945030","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 Average Solution of a TSP Instance in a Graph 图中TSP实例的平均解

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-02-16 DOI: 10.1002/jgt.23232

Stijn Cambie

{"title":"The Average Solution of a TSP Instance in a Graph","authors":"Stijn Cambie","doi":"10.1002/jgt.23232","DOIUrl":"https://doi.org/10.1002/jgt.23232","url":null,"abstract":"<div>\u0000 \u0000 We define the average <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-TSP distance <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>μ</mi>\u0000 \u0000 <mrow>\u0000 <mtext>tsp</mtext>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> as the average length of a shortest closed walk visiting <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> vertices, that is, the expected length of the solution for a random TSP instance with <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> uniformly random chosen vertices. We prove relations with the average <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>μ</mi>\u0000 \u0000 <mrow>\u0000 <mtext>tsp</mtext>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\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> given the order of the graph.\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"332-338"},"PeriodicalIF":0.9,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945031","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

Independent Sets of Random Trees and Sparse Random Graphs

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-02-10 DOI: 10.1002/jgt.23225

Steven Heilman

{"title":"Independent Sets of Random Trees and Sparse Random Graphs","authors":"Steven Heilman","doi":"10.1002/jgt.23225","DOIUrl":"https://doi.org/10.1002/jgt.23225","url":null,"abstract":"An independent set of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in a finite undirected graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> vertices of the graph, no two of which are connected by an edge. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>x</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> be the number of independent sets of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in the graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>α</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>max</mi>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"294-309"},"PeriodicalIF":0.9,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23225","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944734","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

Refining Tree-Decompositions so That They Display the k-Blocks

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-02-10 DOI: 10.1002/jgt.23230

Sandra Albrechtsen

{"title":"Refining Tree-Decompositions so That They Display the k-Blocks","authors":"Sandra Albrechtsen","doi":"10.1002/jgt.23230","DOIUrl":"https://doi.org/10.1002/jgt.23230","url":null,"abstract":"Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>V</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of adhesion less than <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> that efficiently distinguishes every two distinct <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-profiles, and which has the further property that every separable <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-block is equal to the unique part of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>V</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in which it is contained. We give a shorter proof of this result by showing that such a tree-decomposition can in fact be obtained from any canonical tight tree-decomposition of adhesion less than <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. For this, we decompose the parts of such a tree-decomposition by further tree-decompositions. As an application, we also obtain a generalization of Carmesin and Gollin's result to locally finite graphs.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"310-314"},"PeriodicalIF":0.9,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23230","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944735","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

Directed Graphs Without Rainbow Triangles

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-02-04 DOI: 10.1002/jgt.23224

Sebastian Babiński, Andrzej Grzesik, Magdalena Prorok

{"title":"Directed Graphs Without Rainbow Triangles","authors":"Sebastian Babiński, Andrzej Grzesik, Magdalena Prorok","doi":"10.1002/jgt.23224","DOIUrl":"https://doi.org/10.1002/jgt.23224","url":null,"abstract":"<div>\u0000 \u0000 One of the most fundamental results in graph theory is Mantel's theorem which determines the maximum number of edges in a triangle-free graph of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Recently, a colorful variant of this problem has been solved. In this variant we consider <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> graphs on a common vertex set, think of each graph as edges in a distinct color, and want to determine the smallest number of edges in each color which guarantees the existence of a rainbow triangle. Here, we solve the analogous problem for directed graphs without rainbow triangles, either directed or transitive, for any number of colors. The constructions and proofs essentially differ for <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and the type of the forbidden triangle. Additionally, we also solve the analogous problem in the setting of oriented graphs.\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"269-281"},"PeriodicalIF":0.9,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944668","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

Polynomial Characterizations of Distance-Biregular Graphs

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-02-04 DOI: 10.1002/jgt.23227

Sabrina Lato

引用次数: 0