Journal of Graph Theory最新文献_第9页

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

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-14 DOI: 10.1002/jgt.23147

Benedek Kovács, Zoltán Lóránt Nagy

{"title":"Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles","authors":"Benedek Kovács, Zoltán Lóránt Nagy","doi":"10.1002/jgt.23147","DOIUrl":"10.1002/jgt.23147","url":null,"abstract":"In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>ex</mtext>\u0000 \u0000 <mi>F</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${text{ex}}_{F}(n,G)$</annotation>\u0000 </semantics></math> denote the maximum number of edge-disjoint copies of a fixed simple graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> that can be placed on an <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex ground set without forming a subgraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> whose edges are from different <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>-copies. The case when both <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"629-641"},"PeriodicalIF":0.9,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141649705","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

Thin edges in cubic braces 立方括号中的细边

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-14 DOI: 10.1002/jgt.23150

Xiaoling He, Fuliang Lu

{"title":"Thin edges in cubic braces","authors":"Xiaoling He, Fuliang Lu","doi":"10.1002/jgt.23150","DOIUrl":"10.1002/jgt.23150","url":null,"abstract":"For a vertex set <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> in a graph, the edge cut <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∂</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>X</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $partial (X)$</annotation>\u0000 </semantics></math> is the set of edges with exactly one end vertex in <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math>. An edge cut <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∂</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>X</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $partial (X)$</annotation>\u0000 </semantics></math> is tight if every perfect matching of the graph contains exactly one edge in <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∂</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>X</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $partial (X)$</annotation>\u0000 </semantics></math>. A matching covered bipartite graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a brace if, for every tight cut <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∂</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>X</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"642-675"},"PeriodicalIF":0.9,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141650079","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

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

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-11 DOI: 10.1002/jgt.23149

Xihe Li, Hajo Broersma, Ligong Wang

{"title":"Growth rates of the bipartite Erdős–Gyárfás function","authors":"Xihe Li, Hajo Broersma, Ligong Wang","doi":"10.1002/jgt.23149","DOIUrl":"10.1002/jgt.23149","url":null,"abstract":"Given two graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G,H$</annotation>\u0000 </semantics></math> and a positive integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>, an <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(H,q)$</annotation>\u0000 </semantics></math>-coloring of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is an edge-coloring of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that every copy of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> in <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> receives at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"597-628"},"PeriodicalIF":0.9,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23149","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141613046","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

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

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-09 DOI: 10.1002/jgt.23148

Tao Wang, Baoyindureng Wu

{"title":"Maximum odd induced subgraph of a graph concerning its chromatic number","authors":"Tao Wang, Baoyindureng Wu","doi":"10.1002/jgt.23148","DOIUrl":"10.1002/jgt.23148","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>o</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 <annotation> ${f}_{o}(G)$</annotation>\u0000 </semantics></math> be the maximum order of an odd induced subgraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. In 1992, Scott proposed a conjecture that <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>o</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mfrac>\u0000 <mi>n</mi>\u0000 \u0000 <mrow>\u0000 <mi>χ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${f}_{o}(G)ge frac{n}{chi (G)}$</annotation>\u0000 </semantics></math> for a graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> without isolated vertices, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>χ</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"578-596"},"PeriodicalIF":0.9,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141571531","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

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

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-02 DOI: 10.1002/jgt.23125

Daniel W. Cranston, Gexin Yu

{"title":"Cliques in squares of graphs with maximum average degree less than 4","authors":"Daniel W. Cranston, Gexin Yu","doi":"10.1002/jgt.23125","DOIUrl":"10.1002/jgt.23125","url":null,"abstract":"Hocquard, Kim, and Pierron constructed, for every even integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $Dge 2$</annotation>\u0000 </semantics></math>, a 2-degenerate graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>D</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${G}_{D}$</annotation>\u0000 </semantics></math> with maximum degree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ω</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msubsup>\u0000 <mi>G</mi>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msubsup>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mfrac>\u0000 <mn>5</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 \u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $omega ({G}_{D}^{2})=frac{5}{2}D$</annotation>\u0000 </semantics></math>. We prove for (a) all 2-degenerate graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> and (b) all graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"559-577"},"PeriodicalIF":0.9,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23125","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A characterization of regular partial cubes whose all convex cycles have the same lengths 所有凸循环长度相同的规则局部立方体的特征描述

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-28 DOI: 10.1002/jgt.23126

Yan-Ting Xie, Yong-De Feng, Shou-Jun Xu

{"title":"A characterization of regular partial cubes whose all convex cycles have the same lengths","authors":"Yan-Ting Xie, Yong-De Feng, Shou-Jun Xu","doi":"10.1002/jgt.23126","DOIUrl":"10.1002/jgt.23126","url":null,"abstract":"Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $2n$</annotation>\u0000 </semantics></math> where <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>⩾</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ngeqslant 4$</annotation>\u0000 </semantics></math>) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $2n$</annotation>\u0000 </semantics></math>-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder—regular median graphs are hypercubes.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"550-558"},"PeriodicalIF":0.9,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504322","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 bounds for chromatic number VIII. Excluding a path and a complete multipartite graph 色度数的多项式边界 VIII.排除路径和完整多方图

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-24 DOI: 10.1002/jgt.23129

Tung Nguyen, Alex Scott, Paul Seymour

{"title":"Polynomial bounds for chromatic number VIII. Excluding a path and a complete multipartite graph","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"10.1002/jgt.23129","DOIUrl":"10.1002/jgt.23129","url":null,"abstract":"We prove that for every path <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math>, and every integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>, there is a polynomial <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $f$</annotation>\u0000 </semantics></math> such that every graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with chromatic number greater than <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>t</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $f(t)$</annotation>\u0000 </semantics></math> either contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> as an induced subgraph, or contains as a subgraph the complete <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>-partite graph with parts of cardinality <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>. For <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"509-521"},"PeriodicalIF":0.9,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23129","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504375","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

Attainable bounds for algebraic connectivity and maximally connected regular graphs 代数连通性和最大连通正则图的可实现边界

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-24 DOI: 10.1002/jgt.23146

Geoffrey Exoo, Theodore Kolokolnikov, Jeanette Janssen, Timothy Salamon

{"title":"Attainable bounds for algebraic connectivity and maximally connected regular graphs","authors":"Geoffrey Exoo, Theodore Kolokolnikov, Jeanette Janssen, Timothy Salamon","doi":"10.1002/jgt.23146","DOIUrl":"10.1002/jgt.23146","url":null,"abstract":"We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon–Boppana–Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for well-known special cases. For the diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> up to and including <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>9</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=9$</annotation>\u0000 </semantics></math> (the case of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>10</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=10$</annotation>\u0000 </semantics></math> is open). These graphs are extremely rare and also have high girth; for example, we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>7</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D=7$</annotation>\u0000 </semantics></math>; all have girth 8. We also exhibit several infinite families attaining the upper bound with respect to diameter or girth. In particular, when <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math> is a power of prime, we construct a <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"522-549"},"PeriodicalIF":0.9,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23146","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504369","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 oriented m $m$ -semiregular representations of finite groups 关于有限群的定向 m $m$ 半圆代表

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-21 DOI: 10.1002/jgt.23145

Jia-Li Du, Yan-Quan Feng, Sejeong Bang

{"title":"On oriented \u0000 \u0000 \u0000 \u0000 m\u0000 \u0000 \u0000 $m$\u0000 -semiregular representations of finite groups","authors":"Jia-Li Du, Yan-Quan Feng, Sejeong Bang","doi":"10.1002/jgt.23145","DOIUrl":"10.1002/jgt.23145","url":null,"abstract":"A finite group <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> admits an oriented regular representation if there exists a Cayley digraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> such that it has no digons and its automorphism group is isomorphic to <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-semiregular representations using <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-Cayley digraphs. Given a finite group <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, an <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>-Cayley digraph of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a digraph that has a group of automorphisms isomorphic to <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"485-508"},"PeriodicalIF":0.9,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504370","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

Highly connected triples and Mader's conjecture 高连接三元组和马德猜想

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-06-17 DOI: 10.1002/jgt.23144

Qinghai Liu, Kai Ying, Yanmei Hong

{"title":"Highly connected triples and Mader's conjecture","authors":"Qinghai Liu, Kai Ying, Yanmei Hong","doi":"10.1002/jgt.23144","DOIUrl":"https://doi.org/10.1002/jgt.23144","url":null,"abstract":"Mader proved that, for any tree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $T$</annotation>\u0000 </semantics></math> of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math>, every <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-connected graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> with <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 <mn>2</mn>\u0000 \u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $delta (G)ge 2{(k+m-1)}^{2}+m-1$</annotation>\u0000 </semantics></math> contains a subtree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>T</mi>\u0000 \u0000 <mo>′</mo>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"478-484"},"PeriodicalIF":0.9,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142234910","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