Journal of Graph Theory最新文献_第9页

Strong arc decompositions of split digraphs 分裂图的强弧分解

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-08-06 DOI: 10.1002/jgt.23157

Jørgen Bang-Jensen, Yun Wang

{"title":"Strong arc decompositions of split digraphs","authors":"Jørgen Bang-Jensen, Yun Wang","doi":"10.1002/jgt.23157","DOIUrl":"10.1002/jgt.23157","url":null,"abstract":"A strong arc decomposition of a digraph <math>\u0000 \u0000 <mrow>\u0000 <mi>D</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>A</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is a partition of its arc set <math>\u0000 \u0000 <mrow>\u0000 <mi>A</mi>\u0000 </mrow></math> into two sets <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow></math> such that the digraph <math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>V</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>A</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is strong for <math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow></math>. Bang-Jensen and Yeo conjectured that there is some <math>\u0000 \u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow></math> such that every <math>\u0000 \u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow></math>-arc-strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang extended this result to locally semicomplete digraphs by proving that every 2-arc-strong locally semicomplete digraph which is not the square of an even cycle h","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 1","pages":"5-26"},"PeriodicalIF":0.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23157","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943301","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

Stability of Rose Window graphs 玫瑰窗图形的稳定性

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-08-05 DOI: 10.1002/jgt.23162

Milad Ahanjideh, István Kovács, Klavdija Kutnar

{"title":"Stability of Rose Window graphs","authors":"Milad Ahanjideh, István Kovács, Klavdija Kutnar","doi":"10.1002/jgt.23162","DOIUrl":"10.1002/jgt.23162","url":null,"abstract":"A graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}$</annotation>\u0000 </semantics></math> is said to be stable if for the direct product <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mtext>Aut</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Gamma }}times {{bf{K}}}_{2},text{Aut}({rm{Gamma }}times {{bf{K}}}_{2})$</annotation>\u0000 </semantics></math> is isomorphic to <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>Aut</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>Γ</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <msub>\u0000 <mi>Z</mi>\u0000 \u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{Aut}({rm{Gamma }})times {{mathbb{Z}}}_{2}$</annotation>\u0000 </semantics></math>; otherwise, it is called unstable. An unstable graph is called nontrivially unstable when it is not bipartite and no two vertices have the same neighborhood. Wilson described nine families of unstable Rose Window graphs and conjectured that these contain all nontrivially unstable Rose Window graphs (2008). In this paper we show that the conjecture is true.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"810-832"},"PeriodicalIF":0.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943338","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

Proper edge colorings of planar graphs with rainbow C 4 ${C}_{4}$ -s 具有彩虹 C4 ${C}_{4}$-s 的平面图的适当边着色

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-08-05 DOI: 10.1002/jgt.23163

András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy

{"title":"Proper edge colorings of planar graphs with rainbow \u0000 \u0000 \u0000 \u0000 \u0000 C\u0000 4\u0000 \u0000 \u0000 \u0000 ${C}_{4}$\u0000 -s","authors":"András Gyárfás, Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy","doi":"10.1002/jgt.23163","DOIUrl":"10.1002/jgt.23163","url":null,"abstract":"We call a proper edge coloring of 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> a B-coloring if every 4-cycle 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 colored with four different colors. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>q</mi>\u0000 \u0000 <mi>B</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> ${q}_{B}(G)$</annotation>\u0000 </semantics></math> denote the smallest number of colors needed for a B-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>. Motivated by earlier papers on B-colorings, here we consider <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>q</mi>\u0000 \u0000 <mi>B</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> ${q}_{B}(G)$</annotation>\u0000 </semantics></math> for planar and outerplanar graphs in terms of the maximum degree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>Δ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{Delta }}={rm{Delta }}(G)$</annotation>\u0000 </sem","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"833-846"},"PeriodicalIF":0.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943351","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

Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs” 对 "在随机几何图中嵌入平衡生成树的锐阈值 "的更正

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-31 DOI: 10.1002/jgt.23160

{"title":"Correction to “Sharp threshold for embedding balanced spanning trees in random geometric graphs”","authors":"","doi":"10.1002/jgt.23160","DOIUrl":"10.1002/jgt.23160","url":null,"abstract":"A. Espuny Díaz, L. Lichev, D. Mitsche, and A. Wesolek, Sharp threshold for embedding balanced spanning trees in random geometric graphs, J. Graph Theory. 107 (2024), 107–125. https://doi.org/10.1002/jgt.23106In the “ACKNOWLEDGMENTS” section, the text “The research leading to these results has been supported by the Carl-Zeiss-Foundation (Alberto Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002, and by Fondecyt grant 1220174 (Dieter Mitsche) and by the Vanier Scholarship Program (Alexandra Wesolek).” was incorrect.This should have read: “The research leading to these results has been supported by the Carl-Zeiss-Foundation and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project no. 447645533 (A. Espuny Díaz), by grant GrHyDy ANR-20-CE40-0002 and by Fondecyt grant 1220174 (D. Mitsche) and by the Vanier Scholarship Program (A. Wesolek).”The new Funding Information should read as follows:Carl-Zeiss-Foundation.Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project no. 447645533.Grant GrHyDy ANR-20-CE40-0002.Fondecyt, Grant/Award Number: 1220174.The Vanier Scholarship Program.We apologize for this error.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"847"},"PeriodicalIF":0.9,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866312","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 the minimum number of arcs in 4-dicritical oriented graphs 论四临界定向图中的最小弧数

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-29 DOI: 10.1002/jgt.23159

Frédéric Havet, Lucas Picasarri-Arrieta, Clément Rambaud

{"title":"On the minimum number of arcs in 4-dicritical oriented graphs","authors":"Frédéric Havet, Lucas Picasarri-Arrieta, Clément Rambaud","doi":"10.1002/jgt.23159","DOIUrl":"10.1002/jgt.23159","url":null,"abstract":"The dichromatic number <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mover>\u0000 <mi>χ</mi>\u0000 \u0000 <mo>→</mo>\u0000 </mover>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $overrightarrow{chi }(D)$</annotation>\u0000 </semantics></math> of a digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-dicritical if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mover>\u0000 <mi>χ</mi>\u0000 \u0000 <mo>→</mo>\u0000 </mover>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>D</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $overrightarrow{chi }(D)=k$</annotation>\u0000 </semantics></math> and each proper subdigraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $H$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> satisfies <mat","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"778-809"},"PeriodicalIF":0.9,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866313","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

Counting triangles in regular graphs 计算规则图形中的三角形

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-25 DOI: 10.1002/jgt.23156

Jialin He, Xinmin Hou, Jie Ma, Tianying Xie

{"title":"Counting triangles in regular graphs","authors":"Jialin He, Xinmin Hou, Jie Ma, Tianying Xie","doi":"10.1002/jgt.23156","DOIUrl":"10.1002/jgt.23156","url":null,"abstract":"In this paper, we investigate the minimum number of triangles, denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $t(n,k)$</annotation>\u0000 </semantics></math>, in <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-regular graphs, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is an odd integer and <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> is an even integer. The well-known Andrásfai–Erdős–Sós Theorem has established that <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>></mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <an","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"759-777"},"PeriodicalIF":0.9,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23156","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771226","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

Counting rainbow triangles in edge-colored graphs 计算边色图中的彩虹三角形

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-23 DOI: 10.1002/jgt.23158

Xueliang Li, Bo Ning, Yongtang Shi, Shenggui Zhang

{"title":"Counting rainbow triangles in edge-colored graphs","authors":"Xueliang Li, Bo Ning, Yongtang Shi, Shenggui Zhang","doi":"10.1002/jgt.23158","DOIUrl":"10.1002/jgt.23158","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> be an edge-colored graph on <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices. The minimum color degree of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>δ</mi>\u0000 \u0000 <mi>c</mi>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${delta }^{c}(G)$</annotation>\u0000 </semantics></math>, is defined as the minimum number of colors assigned to the edges incident to a vertex in <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. In 2013, Li proved that an edge-colored graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> on <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> vertices contains a rainbow triangle if <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>δ</mi>\u0000 \u0000 <mi>c</mi>\u0000 </msup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </m","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"742-758"},"PeriodicalIF":0.9,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771227","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

Defective acyclic colorings of planar graphs 平面图形的缺陷非循环着色

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-21 DOI: 10.1002/jgt.23154

On-Hei Solomon Lo, Ben Seamone, Xuding Zhu

{"title":"Defective acyclic colorings of planar graphs","authors":"On-Hei Solomon Lo, Ben Seamone, Xuding Zhu","doi":"10.1002/jgt.23154","DOIUrl":"10.1002/jgt.23154","url":null,"abstract":"This paper studies two variants of defective acyclic coloring of planar graphs. 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> and a coloring <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>φ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $varphi $</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, a 2-colored cycle (2CC) transversal is a subset <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>E</mi>\u0000 \u0000 <mo>′</mo>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${E}^{^{prime} }$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>E</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $E(G)$</annotation>\u0000 </semantics></math> that intersects every 2-colored cycle. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> be a positive integer. We denote by <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>m</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 <annotation> ${m}_{k}(G)$</annotation>\u0000 </semantics></math> the minimum integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"729-741"},"PeriodicalIF":0.9,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744614","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

Cycles in 3-connected claw-free planar graphs and 4-connected planar graphs without 4-cycles 3连通无爪平面图和4连通无4周期平面图中的周期

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-21 DOI: 10.1002/jgt.23152

On-Hei Solomon Lo

{"title":"Cycles in 3-connected claw-free planar graphs and 4-connected planar graphs without 4-cycles","authors":"On-Hei Solomon Lo","doi":"10.1002/jgt.23152","DOIUrl":"10.1002/jgt.23152","url":null,"abstract":"The cycle spectrum <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>CS</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{CS}}(G)$</annotation>\u0000 </semantics></math> of 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> is the set of the cycle lengths in <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>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{G}}$</annotation>\u0000 </semantics></math> be a graph class. For any integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $kge 3$</annotation>\u0000 </semantics></math>, define <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>f</mi>\u0000 \u0000 <mi>G</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${f}_{{mathscr{G}}}(k)$</annotation>\u0000 </semantics></math> to be the least integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>k</mi>\u0000 \u0000 <mo>′</mo>\u0000 </msup>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${k}^{^{prime} }ge k$</annotation>\u0000 </semantics>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"702-728"},"PeriodicalIF":0.9,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744613","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

Local version of Vizing's theorem for multigraphs 多图维京定理的局部版本

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-07-16 DOI: 10.1002/jgt.23155

Clinton T. Conley, Jan Grebík, Oleg Pikhurko

{"title":"Local version of Vizing's theorem for multigraphs","authors":"Clinton T. Conley, Jan Grebík, Oleg Pikhurko","doi":"10.1002/jgt.23155","DOIUrl":"10.1002/jgt.23155","url":null,"abstract":"Extending a result of Christiansen, we prove that every multigraph <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 <annotation> $G=(V,E)$</annotation>\u0000 </semantics></math> admits a proper edge colouring <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ϕ</mi>\u0000 \u0000 <mo>:</mo>\u0000 \u0000 <mi>E</mi>\u0000 \u0000 <mo>→</mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <mn>1</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mo>…</mo>\u0000 <mspace></mspace>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $phi :Eto {1,2,ldots ,}$</annotation>\u0000 </semantics></math> which is local, that is, <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ϕ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>e</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>d</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"693-701"},"PeriodicalIF":0.9,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23155","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721055","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