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Computing the Number and Average Size of Connected Sets in Planar 3-Trees 计算平面 3 树中连接集的数量和平均大小
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-17 DOI: 10.1007/s00373-024-02783-8
Zuwen Luo, Kexiang Xu
{"title":"Computing the Number and Average Size of Connected Sets in Planar 3-Trees","authors":"Zuwen Luo, Kexiang Xu","doi":"10.1007/s00373-024-02783-8","DOIUrl":"https://doi.org/10.1007/s00373-024-02783-8","url":null,"abstract":"<p>A vertex set in a graph is a connected set if it induces a connected subgraph. For a tree <i>T</i>, each subgraph induced by a connected set of <i>T</i> is actually a subtree of <i>T</i>. The number and average size of subtrees of a tree <i>T</i> are two well-studied parameters. Yan and Yeh developed a linear-time algorithm for computing the number of subtrees in a tree through “generating function”. In this paper, we present linear-time algorithms for computing the number and average size of connected sets in a planar 3-tree.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140616153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Semi-strict Chordality of Digraphs 数形的半严格弦性
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-17 DOI: 10.1007/s00373-024-02778-5
Jing Huang, Ying Ying Ye
{"title":"Semi-strict Chordality of Digraphs","authors":"Jing Huang, Ying Ying Ye","doi":"10.1007/s00373-024-02778-5","DOIUrl":"https://doi.org/10.1007/s00373-024-02778-5","url":null,"abstract":"<p>Chordal graphs are important in structural graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. Unlike chordal graphs, chordal digraphs lack many structural properties such as forbidden subdigraph or representation characterizations. In this paper we introduce the notion of semi-strict chordal digraphs which form a class strictly between chordal digraphs and chordal graphs. Semi-strict chordal digraphs have rich structural properties. We characterize semi-strict chordal digraphs in terms of knotting graphs, a notion analogous to the one introduced by Gallai for the study of comparability graphs. We also give forbidden subdigraph characterizations of semi-strict chordal digraphs within the classes of locally semicomplete digraphs and weakly quasi-transitive digraphs.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"200 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140616155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Gallai–Ramsey Multiplicity 加莱-拉姆齐多重性
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-14 DOI: 10.1007/s00373-024-02780-x
Yaping Mao
{"title":"Gallai–Ramsey Multiplicity","authors":"Yaping Mao","doi":"10.1007/s00373-024-02780-x","DOIUrl":"https://doi.org/10.1007/s00373-024-02780-x","url":null,"abstract":"<p>Given two graphs <i>G</i> and <i>H</i>, the <i>general</i> <i>k</i><i>-colored Gallai–Ramsey number</i> <span>({text {gr}}_k(G:H))</span> is defined to be the minimum integer <i>m</i> such that every <i>k</i>-coloring of the complete graph on <i>m</i> vertices contains either a rainbow copy of <i>G</i> or a monochromatic copy of <i>H</i>. Interesting problems arise when one asks how many such rainbow copy of <i>G</i> and monochromatic copy of <i>H</i> must occur. The <i>Gallai–Ramsey multiplicity</i> <span>({text {GM}}_{k}(G:H))</span> is defined as the minimum total number of rainbow copy of <i>G</i> and monochromatic copy of <i>H</i> in any exact <i>k</i>-coloring of <span>(K_{{text {gr}}_{k}(G:H)})</span>. In this paper, we give upper and lower bounds for Gallai–Ramsey multiplicity involving some small rainbow subgraphs.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"87 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Minimal Obstructions for Polarity, Monopolarity, Unipolarity and (s, 1)-Polarity in Generalizations of Cographs 极性、单极性、单极性和 (s, 1)- 极性在 Cographs 广义中的最小障碍
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-09 DOI: 10.1007/s00373-024-02784-7
Fernando Esteban Contreras-Mendoza, César Hernández-Cruz
{"title":"Minimal Obstructions for Polarity, Monopolarity, Unipolarity and (s, 1)-Polarity in Generalizations of Cographs","authors":"Fernando Esteban Contreras-Mendoza, César Hernández-Cruz","doi":"10.1007/s00373-024-02784-7","DOIUrl":"https://doi.org/10.1007/s00373-024-02784-7","url":null,"abstract":"<p>It is known that every hereditary property can be characterized by finitely many minimal obstructions when restricted to either the class of cographs or the class of <span>(P_4)</span>-reducible graphs. In this work, we prove that the same is true when restricted to some other superclasses of cographs, including <span>(P_4)</span>-sparse and <span>(P_4)</span>-extendible graphs (both of which extend <span>(P_4)</span>-reducible graphs). We also present complete lists of <span>(P_4)</span>-sparse and <span>(P_4)</span>-extendible minimal obstructions for polarity, monopolarity, unipolarity, and (<i>s</i>, 1)-polarity, where <i>s</i> is a positive integer. In parallel to the case of <span>(P_4)</span>-reducible graphs, all the <span>(P_4)</span>-sparse minimal obstructions for these hereditary properties are cographs.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On the Existence of Small Strictly Neumaier Graphs 论小型严格诺伊迈尔图的存在
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-08 DOI: 10.1007/s00373-024-02779-4
Aida Abiad, Maarten De Boeck, Sjanne Zeijlemaker
{"title":"On the Existence of Small Strictly Neumaier Graphs","authors":"Aida Abiad, Maarten De Boeck, Sjanne Zeijlemaker","doi":"10.1007/s00373-024-02779-4","DOIUrl":"https://doi.org/10.1007/s00373-024-02779-4","url":null,"abstract":"<p>A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters (16, 9, 4; 2, 4), we establish the existence of a strictly Neumaier graph with parameters (25, 12, 5; 2, 5), and we disprove the existence of strictly Neumaier graphs with parameters (25, 16, 9; 3, 5), (28, 18, 11; 4, 7), (33, 24, 17; 6, 9), (35, 2212; 3, 5), (40, 30, 22; 7, 10) and (55, 34, 18; 3, 5). Our proofs use combinatorial techniques and a novel application of integer programming methods.\u0000</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"87 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Stability of Generalized Turán Number for Linear Forests 线性森林广义图兰数的稳定性
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-08 DOI: 10.1007/s00373-024-02781-w
Yisai Xue, Yichong Liu, Liying Kang
{"title":"Stability of Generalized Turán Number for Linear Forests","authors":"Yisai Xue, Yichong Liu, Liying Kang","doi":"10.1007/s00373-024-02781-w","DOIUrl":"https://doi.org/10.1007/s00373-024-02781-w","url":null,"abstract":"<p>Given a graph <i>T</i> and a family of graphs <span>({mathcal {F}})</span>, the generalized Turán number of <span>({mathcal {F}})</span> is the maximum number of copies of <i>T</i> in an <span>({mathcal {F}})</span>-free graph on <i>n</i> vertices, denoted by <span>(ex(n,T,{mathcal {F}}))</span>. A linear forest is a forest whose connected components are all paths and isolated vertices. Let <span>({mathcal {L}}_{k})</span> be the family of all linear forests of size <i>k</i> without isolated vertices. In this paper, we obtained the maximum possible number of <i>r</i>-cliques in <i>G</i>, where <i>G</i> is <span>({mathcal {L}}_{k})</span>-free with minimum degree at least <i>d</i>. Furthermore, we give a stability version of the result. As an application of the stability version of the result, we obtain a clique version of the stability of the Erdős–Gallai Theorem on matchings.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"87 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 Matrix for Counting Paths in Acyclic Colored Digraphs 无环彩色图中的路径计数矩阵
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-08 DOI: 10.1007/s00373-024-02785-6
Sudip Bera
{"title":"A Matrix for Counting Paths in Acyclic Colored Digraphs","authors":"Sudip Bera","doi":"10.1007/s00373-024-02785-6","DOIUrl":"https://doi.org/10.1007/s00373-024-02785-6","url":null,"abstract":"<p>In this paper, we generalize a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"268 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 primality graph of critical 3-hypergraphs 临界 3-hypergraph 的基元图
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-06 DOI: 10.1007/s00373-024-02772-x
{"title":"The primality graph of critical 3-hypergraphs","authors":"","doi":"10.1007/s00373-024-02772-x","DOIUrl":"https://doi.org/10.1007/s00373-024-02772-x","url":null,"abstract":"<h3>Abstract</h3> <p>Given a 3-hypergraph <em>H</em>, a subset <em>M</em> of <em>V</em>(<em>H</em>) is a module of <em>H</em> if for each <span> <span>(ein E(H))</span> </span> such that <span> <span>(ecap Mne emptyset )</span> </span> and <span> <span>(e{setminus } Mne emptyset )</span> </span>, there exists <span> <span>(min M)</span> </span> such that <span> <span>(ecap M={m})</span> </span> and for every <span> <span>(nin M)</span> </span>, we have <span> <span>((e{setminus }{m})cup {n}in E(H))</span> </span>. For example, <span> <span>(emptyset )</span> </span>, <em>V</em>(<em>H</em>) and <span> <span>({v})</span> </span>, where <span> <span>(vin V(H))</span> </span>, are modules of <em>H</em>, called trivial. A 3-hypergraph is prime if all its modules are trivial. Furthermore, a prime 3-hypergraph is critical if all its induced subhypergraphs, obtained by removing one vertex, are not prime. Lastly, we associate with a prime 3-hypergraph its primality graph the edges of which are the unordered pairs of vertices whose removal provides a prime induced subhypergraph. We characterize the critical 3-hypergraphs together with their primality graph.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"87 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On the Edge-Erdős–Pósa Property of Ladders 关于梯子的边缘-厄尔多斯-波萨特性
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-05 DOI: 10.1007/s00373-024-02765-w
Raphael Steck, Arthur Ulmer
{"title":"On the Edge-Erdős–Pósa Property of Ladders","authors":"Raphael Steck, Arthur Ulmer","doi":"10.1007/s00373-024-02765-w","DOIUrl":"https://doi.org/10.1007/s00373-024-02765-w","url":null,"abstract":"<p>We prove that the ladder with 3 rungs and the house graph have the edge-Erdős–Pósa property, while ladders with 14 rungs or more have not. Additionally, we prove that the latter bound is optimal in the sense that the only known counterexample graph does not permit a better result.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"98 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Planar Graphs with the Maximum Number of Induced 4-Cycles or 5-Cycles 具有最多诱导 4 周期或 5 周期的平面图形
IF 0.7 4区 数学
Graphs and Combinatorics Pub Date : 2024-04-05 DOI: 10.1007/s00373-024-02773-w
Michael Savery
{"title":"Planar Graphs with the Maximum Number of Induced 4-Cycles or 5-Cycles","authors":"Michael Savery","doi":"10.1007/s00373-024-02773-w","DOIUrl":"https://doi.org/10.1007/s00373-024-02773-w","url":null,"abstract":"<p>For large <i>n</i> we determine exactly the maximum numbers of induced <span>(C_4)</span> and <span>(C_5)</span> subgraphs that a planar graph on <i>n</i> vertices can contain. We show that <span>(K_{2,n-2})</span> uniquely achieves this maximum in the <span>(C_4)</span> case, and we identify the graphs which achieve the maximum in the <span>(C_5)</span> case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"92 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
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