Discrete Mathematics最新文献_第9页

A note on clique immersion of strong product graphs 关于强积图的簇嵌入的说明

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-09-04 DOI: 10.1016/j.disc.2024.114237

Chuanshu Wu, Zijian Deng

{"title":"A note on clique immersion of strong product graphs","authors":"Chuanshu Wu, Zijian Deng","doi":"10.1016/j.disc.2024.114237","DOIUrl":"10.1016/j.disc.2024.114237","url":null,"abstract":"<div>Let <math><mi>G</mi><mo>,</mo><mi>H</mi></math> be graphs, and <math><mi>G</mi><mo>⁎</mo><mi>H</mi></math> represent a specific graph product of G and H. Define <math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> as the largest t for which G contains a <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>-immersion. Collins, Heenehan, and McDonald posed the question: given <math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>t</mi></math> and <math><mi>i</mi><mi>m</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>r</mi></math>, how large can <math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>⁎</mo><mi>H</mi><mo>)</mo></math> be? They conjectured <math><mi>i</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>⁎</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mi>t</mi><mi>r</mi></math> when ⁎ denotes the strong product. In this note, we affirm that the conjecture holds for graphs with certain immersions, in particular when H contains <math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math> as a subgraph. As a consequence we also get an alternative argument for a result of Guyer and McDonald, showing that the line graphs of constant-multiplicity multigraphs satisfy the conjecture originally proposed by Abu-Khzam and Langston.</div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114237"},"PeriodicalIF":0.7,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003686/pdfft?md5=2175b8b68439085105021d9c5e79d193&pid=1-s2.0-S0012365X24003686-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136787","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

Covering the edges of a graph with triangles 用三角形覆盖图形边缘

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-09-03 DOI: 10.1016/j.disc.2024.114226

Csilla Bujtás , Akbar Davoodi , Laihao Ding , Ervin Győri , Zsolt Tuza , Donglei Yang

引用次数: 0

2-Connected spanning subgraphs of circuit graphs 2 电路图的连接跨度子图

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-09-03 DOI: 10.1016/j.disc.2024.114228

Atsuhiro Nakamoto , Kenta Ozeki , Daiki Takahashi

引用次数: 0

Degree powers and number of stars in graphs with a forbidden broom 带禁忌扫帚的图形中的度数幂和星数

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-09-03 DOI: 10.1016/j.disc.2024.114232

Dániel Gerbner

引用次数: 0

Brooks-type theorems for relaxations of square colorings 方形着色松弛的布鲁克斯型定理

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-09-03 DOI: 10.1016/j.disc.2024.114233

Eun-Kyung Cho , Ilkyoo Choi , Hyemin Kwon , Boram Park

{"title":"Brooks-type theorems for relaxations of square colorings","authors":"Eun-Kyung Cho , Ilkyoo Choi , Hyemin Kwon , Boram Park","doi":"10.1016/j.disc.2024.114233","DOIUrl":"10.1016/j.disc.2024.114233","url":null,"abstract":"<div>The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer h, the proper h-conflict-free chromatic number of a graph G, denoted <math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math>, is the minimum k such that G has a proper k-coloring where every vertex v has <math><mi>min</mi><mo>⁡</mo><mo>{</mo><msub><mrow><mi>deg</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>⁡</mo><mo>(</mo><mi>v</mi><mo>)</mo><mo>,</mo><mi>h</mi><mo>}</mo></math> colors appearing exactly once on its neighborhood. Caro, Petruševski, and Škrekovski put forth a Brooks-type conjecture: if G is a graph with <math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>3</mn></math>, then <math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math>. The best known result regarding the conjecture is <math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></math>, which is implied by a result of Pach and Tardos. We improve upon the aforementioned result for all h, and also enlarge the class of graphs for which the conjecture is known to be true.Our main result is the following: for a graph G, if <math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>h</mi><mo>+</mo><mn>2</mn></math>, then <math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mi>h</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math>; this is tight up to the additive term as we explicitly construct infinitely many graphs G with <math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>pcf</mi></mrow><mrow><mi>h</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>h</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo></math>. We also show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. Our main result also improves upon a Brooks-type result for h-dynamic coloring.</div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114233"},"PeriodicalIF":0.7,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003649/pdfft?md5=d91aefd107404a4f8392a7adc9d9507d&pid=1-s2.0-S0012365X24003649-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129960","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

Induced subgraphs and tree decompositions VI. Graphs with 2-cutsets 诱导子图和树分解 VI.带 2 切集的图

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-09-02 DOI: 10.1016/j.disc.2024.114195

Tara Abrishami , Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl

{"title":"Induced subgraphs and tree decompositions VI. Graphs with 2-cutsets","authors":"Tara Abrishami , Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl","doi":"10.1016/j.disc.2024.114195","DOIUrl":"10.1016/j.disc.2024.114195","url":null,"abstract":"<div>This paper continues a series of papers investigating the following question: which hereditary graph classes have bounded treewidth? We call a graph t-clean if it does not contain as an induced subgraph the complete graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>, the complete bipartite graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math>, subdivisions of a <math><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></math>-wall, and line graphs of subdivisions of a <math><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></math>-wall. It is known that graphs with bounded treewidth must be t-clean for some t; however, it is not true that every t-clean graph has bounded treewidth. In this paper, we show that three types of cutsets, namely clique cutsets, 2-cutsets, and 1-joins, interact well with treewidth and with each other, so graphs that are decomposable by these cutsets into basic classes of bounded treewidth have bounded treewidth. We apply this result to two hereditary graph classes, the class of (<math><mi>I</mi><mi>S</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math>, wheel)-free graphs and the class of graphs with no cycle with a unique chord. These classes were previously studied and decomposition theorems were obtained for both classes. Our main results are that t-clean (<math><mi>I</mi><mi>S</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math>, wheel)-free graphs have bounded treewidth and that t-clean graphs with no cycle with a unique chord have bounded treewidth.</div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114195"},"PeriodicalIF":0.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003261/pdfft?md5=e8262a89abc8297f51785b66fc0ac9c4&pid=1-s2.0-S0012365X24003261-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142122205","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

Toroidal Hitomezashi patterns 环状人字形图案

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-08-30 DOI: 10.1016/j.disc.2024.114231

Qiuyu Ren , Shengtong Zhang

引用次数: 0

On the Ramsey number of the double star 关于双星的拉姆齐数

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2024-08-30 DOI: 10.1016/j.disc.2024.114227

Freddy Flores Dubó, Maya Stein

引用次数: 0

Asymptotically good LCD 2-quasi-abelian codes over finite fields 有限域上渐近良好的 LCD 2-类阿贝尔码