arXiv - MATH - Combinatorics最新文献_第3页

On Geometry and Combinatorics of Finite Classical Polar Spaces 论有限经典极空间的几何学和组合学

arXiv - MATH - Combinatorics Pub Date : 2024-09-17 DOI: arxiv-2409.11131

Valentino Smaldore

{"title":"On Geometry and Combinatorics of Finite Classical Polar Spaces","authors":"Valentino Smaldore","doi":"arxiv-2409.11131","DOIUrl":"https://doi.org/arxiv-2409.11131","url":null,"abstract":"Polar spaces over finite fields are fundamental in combinatorial geometry.\u0000The concept of polar space was firstly introduced by F. Veldkamp who gave a\u0000system of 10 axioms in the spirit of Universal Algebra. Later the axioms were\u0000simplified by J. Tits, who introduced the concept of subspaces. Later on, from\u0000the point of view of incidence geometry, axioms of polar spaces were also given\u0000by F. Buekenhout and E. Shult in 1974. The reader can find the three systems of\u0000axioms of polar spaces in Appendix A. Examples of polar spaces are the so\u0000called Finite classical polar spaces, i.e. incidence structures arising from\u0000quadrics, symplectic forms and Hermitian forms, which are in correspondance\u0000with reflexive sesquilinear forms.It is still an open problem to show whether\u0000or not classical polar spaces are the only example of finite polar spaces. Nowadays, some research problems related to finite classical polar space are:\u0000existence of spreads and ovoids; existence of regular systems and $m$-ovoids;\u0000upper or lower bounds on partial spreads and partial ovoids. Moreover, polar\u0000spaces are in relation with combinatorial objects as regular graphs, block\u0000designs and association schemes. In this Ph.D. Thesis we investigate the geometry of finite classical polar\u0000spaces, giving contributions to the above problems. The thesis is organized as\u0000follows. Part I is more focused on the geometric aspects of polar spaces, while\u0000in Part II some combinatorial objects are introduced such as regular graphs,\u0000association schemes and combinatorial designs. Finally Appendix B, C and D are\u0000dedicated to give more details on, respectively, maximal curves, linear codes\u0000and combinatorial designs, giving useful results and definitions.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"2020 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Complexity of Maximizing the MST-ratio 最大化 MST 比率的复杂性

arXiv - MATH - Combinatorics Pub Date : 2024-09-17 DOI: arxiv-2409.11079

Afrouz Jabal Ameli, Faezeh Motiei, Morteza Saghafian

引用次数: 0

Cops against a cheating robber 警察打击作弊劫匪

arXiv - MATH - Combinatorics Pub Date : 2024-09-17 DOI: arxiv-2409.11581

Nancy E. Clarke, Danny Dyer, William Kellough

引用次数: 0

Algorithmic methods of finite discrete structures. Hamiltonian cycle of a complete graph and the Traveling salesman problem 有限离散结构的算法方法。完整图的哈密顿循环和旅行推销员问题

arXiv - MATH - Combinatorics Pub Date : 2024-09-17 DOI: arxiv-2409.11563

Sergey Kurapov, Maxim Davidovsky, Svetlana Polyuga

引用次数: 0

Backbone coloring for graphs with degree 4 阶数为 4 的图的骨干着色

arXiv - MATH - Combinatorics Pub Date : 2024-09-16 DOI: arxiv-2409.10201

Krzysztof Michalik, Krzysztof Turowski

引用次数: 0

The minimum size of a chordal graph with given order and minimum degree 给定阶数和最小阶数的弦图的最小尺寸

arXiv - MATH - Combinatorics Pub Date : 2024-09-16 DOI: arxiv-2409.10261

Xingzhi Zhan, Leilei Zhang

引用次数: 0

Perfect coalition in graphs 图中的完美联盟

arXiv - MATH - Combinatorics Pub Date : 2024-09-16 DOI: arxiv-2409.10185

Doost Ali Mojdeh, Mohammad Reza Samadzadeh

{"title":"Perfect coalition in graphs","authors":"Doost Ali Mojdeh, Mohammad Reza Samadzadeh","doi":"arxiv-2409.10185","DOIUrl":"https://doi.org/arxiv-2409.10185","url":null,"abstract":"noindent A perfect dominating set in a graph $G=(V,E)$ is a subset $S\u0000subseteq V$ such that each vertex in $V setminus S$ has exactly one neighbor\u0000in $S$. A perfect coalition in $G$ consists of two disjoint sets of vertices\u0000$V_i$ and $V_j$ such that i) neither $V_i$ nor $V_j$ is a dominating set, ii)\u0000each vertex in $V(G) setminus V_i$ has at most one neighbor in $V_i$ and each\u0000vertex in $V(G) setminus V_j$ has at most one neighbor in $V_j$, and iii) $V_i\u0000cup V_j$ is a perfect dominating set. A perfect coalition partition\u0000(abbreviated $prc$-partition) in a graph $G$ is a vertex partition $pi=\u0000lbrace V_1,V_2,dots ,V_k rbrace$ such that for each set $V_i$ of $pi$\u0000either $V_i$ is a singleton dominating set, or there exists a set $V_j in pi$\u0000that forms a perfect coalition with $V_i$. In this paper, we initiate the study\u0000of perfect coalition partitions in graphs. We obtain a bound on the number of\u0000perfect coalitions involving each member of a perfect coalition partition, in\u0000terms of maximum degree. The perfect coalition of some special graphs are\u0000investigated. The graph $G$ with $delta(G)=1$, the triangle-free graphs $G$\u0000with prefect coalition number of order of $G$ and the trees $T$ with prefect\u0000coalition number in ${n,n-1,n-2}$ where $n=|V(T)|$ are characterized.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The maximum size of a nonhamiltonian-connected graph with given order and minimum degree 给定阶数和最小阶数的非哈密顿连接图的最大尺寸

arXiv - MATH - Combinatorics Pub Date : 2024-09-16 DOI: arxiv-2409.10255

Leilei Zhang

引用次数: 0

The Lower Bound for Number of Hexagons in Strongly Regular Graphs with Parameters $λ=1$ and $μ=2$ 参数为 $λ=1$ 和 $μ=2$ 的强正则图中六角形数量的下限

arXiv - MATH - Combinatorics Pub Date : 2024-09-16 DOI: arxiv-2409.10620

Reimbay Reimbayev

引用次数: 0

The strong Nash-Williams orientation theorem for rayless graphs 无射线图的强纳什-威廉斯定向定理

arXiv - MATH - Combinatorics Pub Date : 2024-09-16 DOI: arxiv-2409.10378

Max Pitz, Jacob Stegemann

引用次数: 0