Journal of Graph Theory最新文献_第4页

Packing Colourings in Complete Bipartite Graphs and the Inverse Problem for Correspondence Packing 完全二部图中的填充着色及对应填充的反问题

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-01-06 DOI: 10.1002/jgt.23215

Stijn Cambie, Rimma Hämäläinen

{"title":"Packing Colourings in Complete Bipartite Graphs and the Inverse Problem for Correspondence Packing","authors":"Stijn Cambie, Rimma Hämäläinen","doi":"10.1002/jgt.23215","DOIUrl":"https://doi.org/10.1002/jgt.23215","url":null,"abstract":"<div>\u0000 \u0000 Applications of graph colouring often involve taking restrictions into account, and it is desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition into disjoint colourings, we speak of a packing. However, even for complete bipartite graphs, the list chromatic number can be arbitrarily large, and its exact determination is generally difficult. For the packing variant, this question becomes even harder. In this paper, we study the correspondence- and list-packing numbers of (asymmetric) complete bipartite graphs. In the most asymmetric cases, Latin squares come into play. Our results show that every <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>z</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <msup>\u0000 <mi>Z</mi>\u0000 \u0000 <mo>+</mo>\u0000 </msup>\u0000 \u0000 <mo></mo>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mn>3</mn>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23215:jgt23215-math-0001\" wiley:location=\"equation/jgt23215-math-0001.png\"><mrow><mrow><mi>z</mi><mo>unicode{x02208}</mo><msup><mi mathvariant=\"double-struck\">Z</mi><mo>unicode{x0002B}</mo></msup><mo>unicode{x0005C}</mo><mrow><mo class=\"MathClass-open\">{</mo><mn>3</mn><mo class=\"MathClass-close\">}</mo></mrow></mrow></mrow></math></annotation>\u0000 </semantics></math> can be equal to the correspondence packing number of a graph. We disprove a recent conjecture that relates the list packing number and the list flexibility number. Additionally, we improve the threshold functions for the correspondence packing variant.\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"52-61"},"PeriodicalIF":0.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612344","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

Weak Rainbow Saturation Numbers of Graphs 图的弱彩虹饱和数

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2025-01-06 DOI: 10.1002/jgt.23211

Xihe Li, Jie Ma, Tianying Xie

{"title":"Weak Rainbow Saturation Numbers of Graphs","authors":"Xihe Li, Jie Ma, Tianying Xie","doi":"10.1002/jgt.23211","DOIUrl":"https://doi.org/10.1002/jgt.23211","url":null,"abstract":"<div>\u0000 \u0000 For a fixed graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0001\" wiley:location=\"equation/jgt23211-math-0001.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>, we say that an edge-colored graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0002\" wiley:location=\"equation/jgt23211-math-0002.png\"><mrow><mrow><mi>G</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> is weakly <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0003\" wiley:location=\"equation/jgt23211-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-rainbow saturated if there exists an ordering <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>e</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>e</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>e</mi>\u0000 \u0000 <mi>m</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23211:jgt23211-math-0004\" wiley:location=\"equation/jgt23211-math-0004.png\"><mrow><mrow><msub><mi>e</mi><mn>1</mn></msub><mo>,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>,</mo><mo>unicode{","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"35-42"},"PeriodicalIF":0.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612342","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

Positive Co-Degree Turán Number for C5 and C5− 正共度Turán C5和C5−的数量

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-30 DOI: 10.1002/jgt.23206

Zhuo Wu

{"title":"Positive Co-Degree Turán Number for C5 and C5−","authors":"Zhuo Wu","doi":"10.1002/jgt.23206","DOIUrl":"https://doi.org/10.1002/jgt.23206","url":null,"abstract":"The minimum positive co-degree <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msubsup>\u0000 <mi>δ</mi>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>+</mo>\u0000 </msubsup>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0001\" wiley:location=\"equation/jgt23206-math-0001.png\"><mrow><mrow><msubsup><mi>unicode{x003B4}</mi><mrow><mi>r</mi><mo>unicode{x02212}</mo><mn>1</mn></mrow><mo>unicode{x0002B}</mo></msubsup><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\u0000 </semantics></math> of a nonempty <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0002\" wiley:location=\"equation/jgt23206-math-0002.png\"><mrow><mrow><mi>r</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23206:jgt23206-math-0003\" wiley:location=\"equation/jgt23206-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> is the maximum <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt2","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"25-30"},"PeriodicalIF":0.9,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23206","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612490","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

C10 Has Positive Turán Density in the Hypercube C10在超立方体中的密度为正Turán

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-30 DOI: 10.1002/jgt.23217

Alexandr Grebennikov, João Pedro Marciano

{"title":"C10 Has Positive Turán Density in the Hypercube","authors":"Alexandr Grebennikov, João Pedro Marciano","doi":"10.1002/jgt.23217","DOIUrl":"https://doi.org/10.1002/jgt.23217","url":null,"abstract":"<div>\u0000 \u0000 The <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0001\" wiley:location=\"equation/jgt23217-math-0001.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-dimensional hypercube <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>Q</mi>\u0000 \u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0002\" wiley:location=\"equation/jgt23217-math-0002.png\"><mrow><mrow><msub><mi>Q</mi><mi>n</mi></msub></mrow></mrow></math></annotation>\u0000 </semantics></math> is a graph with vertex set <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 \u0000 <mi>n</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0003\" wiley:location=\"equation/jgt23217-math-0003.png\"><mrow><mrow><msup><mrow><mo class=\"MathClass-open\">{</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo class=\"MathClass-close\">}</mo></mrow><mi>n</mi></msup></mrow></mrow></math></annotation>\u0000 </semantics></math> such that there is an edge between two vertices if and only if they differ in exactly one coordinate. For any graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:j","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"31-34"},"PeriodicalIF":0.9,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612491","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

Extremal Problems for a Matching and Any Other Graph 匹配图和其他图的极值问题

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-19 DOI: 10.1002/jgt.23210

Xiutao Zhu, Yaojun Chen

{"title":"Extremal Problems for a Matching and Any Other Graph","authors":"Xiutao Zhu, Yaojun Chen","doi":"10.1002/jgt.23210","DOIUrl":"https://doi.org/10.1002/jgt.23210","url":null,"abstract":"<div>\u0000 \u0000 For a family of graphs <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0001\" wiley:location=\"equation/jgt23210-math-0001.png\"><mrow><mrow><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">unicode{x02131}</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>, a graph is called <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0002\" wiley:location=\"equation/jgt23210-math-0002.png\"><mrow><mrow><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">unicode{x02131}</mi></mrow></mrow></math></annotation>\u0000 </semantics></math>-free if it does not contain any member of <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0003\" wiley:location=\"equation/jgt23210-math-0003.png\"><mrow><mrow><mi class=\"MJX-tex-caligraphic\" mathvariant=\"normal\">unicode{x02131}</mi></mrow></mrow></math></annotation>\u0000 </semantics></math> as a subgraph. The generalized Turán number <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mtext>ex</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>K</mi>\u0000 \u0000 <mi>r</mi>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>ℱ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23210:jgt23210-math-0004\" wiley:location=\"equation/jgt23210-math-0004.png\"&gt","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"19-24"},"PeriodicalIF":0.9,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612533","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

A Construction of a 3/2-Tough Plane Triangulation With No 2-Factor 无2因子的3/2-坚韧平面三角剖分的构造

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-18 DOI: 10.1002/jgt.23209

Songling Shan

{"title":"A Construction of a 3/2-Tough Plane Triangulation With No 2-Factor","authors":"Songling Shan","doi":"10.1002/jgt.23209","DOIUrl":"https://doi.org/10.1002/jgt.23209","url":null,"abstract":"<div>\u0000 \u0000 In 1956, Tutte proved the celebrated theorem that every 4-connected planar graph is Hamiltonian. This result implies that every more than <math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math>-tough planar graph on at least three vertices is Hamiltonian and so has a 2-factor. Owens in 1999 constructed non-Hamiltonian maximal planar graphs of toughness arbitrarily close to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math> and asked whether there exists a maximal non-Hamiltonian planar graph of toughness exactly <math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math>. In fact, the graphs Owens constructed do not even contain a 2-factor. Thus the toughness of exactly <math>\u0000 <semantics>\u0000 <mrow>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>3</mn>\u0000 \u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics>\u0000 </mrow>\u0000 <annotation> $frac{3}{2}$</annotation>\u0000 </semantics></math> is the only case left in asking the existence of 2-factors in tough planar graphs. This question was also asked by Bauer, Broersma, and Schmeichel in a survey. In this paper, we close this gap by cons","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"5-18"},"PeriodicalIF":0.9,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143612531","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

Equitable List Coloring of Planar Graphs With Given Maximum Degree 给定最大度的平面图的公平表着色

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-15 DOI: 10.1002/jgt.23203

H. A. Kierstead, Alexandr Kostochka, Zimu Xiang

{"title":"Equitable List Coloring of Planar Graphs With Given Maximum Degree","authors":"H. A. Kierstead, Alexandr Kostochka, Zimu Xiang","doi":"10.1002/jgt.23203","DOIUrl":"https://doi.org/10.1002/jgt.23203","url":null,"abstract":"If <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math> is a list assignment of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math> colors to each vertex of an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>, then an equitable <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> is a proper coloring of vertices of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> from their lists such that no color is used more than <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⌈</mo>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>/</mo>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <mo>⌉</mo>\u0000 </mrow>\u0000 <annotation> $lceil n/rrceil $</annotation>\u0000 </semantics></math> times. A graph is equitably <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-choosable if it has an equitable <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <annotation> $L$</annotation>\u0000 </semantics></math>-coloring for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-list assignment <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"832-838"},"PeriodicalIF":0.9,"publicationDate":"2024-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143456080","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 Pre- and Post-Positional Semi-Random Graph Processes 关于前置和后置半随机图过程

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-12 DOI: 10.1002/jgt.23202

Pu Gao, Hidde Koerts

{"title":"On the Pre- and Post-Positional Semi-Random Graph Processes","authors":"Pu Gao, Hidde Koerts","doi":"10.1002/jgt.23202","DOIUrl":"https://doi.org/10.1002/jgt.23202","url":null,"abstract":"We study the semi-random graph process, and a variant process recently suggested by Nick Wormald. We show that these two processes are asymptotically equally fast in constructing a semi-random graph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> that has property <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math>, for the following examples of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math>: (1) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math> is the set of graphs containing a fixed <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation> $d$</annotation>\u0000 </semantics></math>-degenerate subgraph, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $dge 1$</annotation>\u0000 </semantics></math> is fixed and (2) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{P}}$</annotation>\u0000 </semantics></math> is the set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-connected graphs, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation> $kge 1$</annotation>\u0000 </semantics></math> is fixed. In particular, our result of the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-connectedness above settles the open case <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"819-831"},"PeriodicalIF":0.9,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23202","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455949","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 a Question of Erdős and Nešetřil About Minimal Cuts in a Graph 关于图中最小割的Erdős和Nešetřil问题

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-08 DOI: 10.1002/jgt.23207

Domagoj Bradač

引用次数: 0

Hypergraph Anti-Ramsey Theorems 超图反拉姆齐定理

IF 0.9 3区数学

Journal of Graph Theory Pub Date : 2024-12-08 DOI: 10.1002/jgt.23204

Xizhi Liu, Jialei Song

{"title":"Hypergraph Anti-Ramsey Theorems","authors":"Xizhi Liu, Jialei Song","doi":"10.1002/jgt.23204","DOIUrl":"https://doi.org/10.1002/jgt.23204","url":null,"abstract":"<div>\u0000 \u0000 The anti-Ramsey number <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mtext>ar</mtext>\u0000 \u0000 <mrow>\u0000 \u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>F</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>\u0000 $text{ar}(n,F)$\u0000</annotation>\u0000 </semantics>\u0000 </math> of an <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation>\u0000 $r$\u0000</annotation>\u0000 </semantics>\u0000 </math>-graph <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation>\u0000 $F$\u0000</annotation>\u0000 </semantics>\u0000 </math> is the minimum number of colors needed to color the complete <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>\u0000 $n$\u0000</annotation>\u0000 </semantics>\u0000 </math>-vertex <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation>\u0000 $r$\u0000</annotation>\u0000 </semantics>\u0000 </math>-graph to ensure the existence of a rainbow copy of <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation>\u0000 $F$\u0000</annotation>\u0000 </semantics>\u0000 </math>. We establish a removal-type result for the anti-Ramsey problem of <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation>\u0000 $F$\u0000</annotation>\u0000 </semantics>\u0000 </math> when <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 <annotation>\u0000 $F$\u0000</annotation>\u0000 </semantics>\u0000 </math> is the expansion of a hypergraph with a smaller uniformity. We present two applications of this result. First, we refine the general bound <math>\u0000 \u0000 <semantics>\u0000 \u0000 <mrow>\u0000 <mtext>ar</mtext>\u0000 \u0000 <mrow>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 4","pages":"808-816"},"PeriodicalIF":0.9,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455744","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