Polar spaces over finite fields are fundamental in combinatorial geometry. The concept of polar space was firstly introduced by F. Veldkamp who gave a system of 10 axioms in the spirit of Universal Algebra. Later the axioms were simplified by J. Tits, who introduced the concept of subspaces. Later on, from the point of view of incidence geometry, axioms of polar spaces were also given by F. Buekenhout and E. Shult in 1974. The reader can find the three systems of axioms of polar spaces in Appendix A. Examples of polar spaces are the so called Finite classical polar spaces, i.e. incidence structures arising from quadrics, symplectic forms and Hermitian forms, which are in correspondance with reflexive sesquilinear forms.It is still an open problem to show whether or not classical polar spaces are the only example of finite polar spaces. Nowadays, some research problems related to finite classical polar space are: existence of spreads and ovoids; existence of regular systems and $m$-ovoids; upper or lower bounds on partial spreads and partial ovoids. Moreover, polar spaces are in relation with combinatorial objects as regular graphs, block designs and association schemes. In this Ph.D. Thesis we investigate the geometry of finite classical polar spaces, giving contributions to the above problems. The thesis is organized as follows. Part I is more focused on the geometric aspects of polar spaces, while in Part II some combinatorial objects are introduced such as regular graphs, association schemes and combinatorial designs. Finally Appendix B, C and D are dedicated to give more details on, respectively, maximal curves, linear codes and combinatorial designs, giving useful results and definitions.
有限域上的极空间是组合几何中的基本概念。极空间的概念最早由 F. Veldkamp 提出，他以普遍代数的精神给出了一个包含 10 条公理的系统。后来，J. Tits 简化了公理，引入了子空间的概念。后来，F. Buekenhout 和 E. Shult 又从入射几何的角度，于 1974 年给出了极空间公理。读者可以在附录 A 中找到极空间的三个公理体系。极性空间的例子是所谓的有限经典极性空间，即由四边形、交折形式和赫米提形式产生的入射结构，它们与反身倍线性形式相对应。目前，与有限经典极空间相关的一些研究问题有：展曲面和卵形曲面的存在性；正则系统和 $m$-ovoids 的存在性；部分展曲面和部分卵形曲面的上界或下界。此外，极空间还与正则图、块设计和关联方案等组合对象有关。在这篇博士论文中，我们研究了有限经典极空间的几何，对上述问题做出了贡献。论文的组织结构如下。第一部分更侧重于极空间的几何方面，第二部分介绍了一些组合对象，如正则图、关联方案和组合设计。最后，附录 B、C 和 D 分别详细介绍了最大曲线、线性编码和组合设计，并给出了有用的结果和定义。
{"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}
Given a finite set of red and blue points in $mathbb{R}^d$, the MST-ratio is the combined length of the Euclidean minimum spanning trees of red points and of blue points divided by the length of the Euclidean minimum spanning tree of the union of them. The maximum MST-ratio of a point set is the maximum MST-ratio over all non-trivial colorings of its points by red and blue. We prove that the problem of finding the maximum MST-ratio of a given point set is NP-hard when the dimension is a part of the input. Moreover, we present a $O(n^2)$ running time $3$-approximation algorithm for it. As a part of the proof, we show that in any metric space, the maximum MST-ratio is smaller than $3$. Additionally, we study the average MST-ratio over all colorings of a set of $n$ points. We show that this average is always at least $frac{n-2}{n-1}$, and for $n$ random points uniformly distributed in a $d$-dimensional unit cube, the average tends to $sqrt[d]{2}$ in expectation as $n$ goes to infinity.
{"title":"The Complexity of Maximizing the MST-ratio","authors":"Afrouz Jabal Ameli, Faezeh Motiei, Morteza Saghafian","doi":"arxiv-2409.11079","DOIUrl":"https://doi.org/arxiv-2409.11079","url":null,"abstract":"Given a finite set of red and blue points in $mathbb{R}^d$, the MST-ratio is\u0000the combined length of the Euclidean minimum spanning trees of red points and\u0000of blue points divided by the length of the Euclidean minimum spanning tree of\u0000the union of them. The maximum MST-ratio of a point set is the maximum\u0000MST-ratio over all non-trivial colorings of its points by red and blue. We\u0000prove that the problem of finding the maximum MST-ratio of a given point set is\u0000NP-hard when the dimension is a part of the input. Moreover, we present a\u0000$O(n^2)$ running time $3$-approximation algorithm for it. As a part of the\u0000proof, we show that in any metric space, the maximum MST-ratio is smaller than\u0000$3$. Additionally, we study the average MST-ratio over all colorings of a set\u0000of $n$ points. We show that this average is always at least $frac{n-2}{n-1}$,\u0000and for $n$ random points uniformly distributed in a $d$-dimensional unit cube,\u0000the average tends to $sqrt[d]{2}$ in expectation as $n$ goes to infinity.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255066","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}
We investigate a cheating robot version of Cops and Robber, first introduced by Huggan and Nowakowski, where both the cops and the robber move simultaneously, but the robber is allowed to react to the cops' moves. For conciseness, we refer to this game as Cops and Cheating Robot. The cheating robot number for a graph is the fewest number of cops needed to win on the graph. We introduce a new parameter for this variation, called the push number, which gives the value for the minimum number of cops that move onto the robber's vertex given that there are a cheating robot number of cops on the graph. After producing some elementary results on the push number, we use it to give a relationship between Cops and Cheating Robot and Surrounding Cops and Robbers. We investigate the cheating robot number for planar graphs and give a tight bound for bipartite planar graphs. We show that determining whether a graph has a cheating robot number at most fixed $k$ can be done in polynomial time. We also obtain bounds on the cheating robot number for strong and lexicographic products of graphs.
{"title":"Cops against a cheating robber","authors":"Nancy E. Clarke, Danny Dyer, William Kellough","doi":"arxiv-2409.11581","DOIUrl":"https://doi.org/arxiv-2409.11581","url":null,"abstract":"We investigate a cheating robot version of Cops and Robber, first introduced\u0000by Huggan and Nowakowski, where both the cops and the robber move\u0000simultaneously, but the robber is allowed to react to the cops' moves. For\u0000conciseness, we refer to this game as Cops and Cheating Robot. The cheating\u0000robot number for a graph is the fewest number of cops needed to win on the\u0000graph. We introduce a new parameter for this variation, called the push number,\u0000which gives the value for the minimum number of cops that move onto the\u0000robber's vertex given that there are a cheating robot number of cops on the\u0000graph. After producing some elementary results on the push number, we use it to\u0000give a relationship between Cops and Cheating Robot and Surrounding Cops and\u0000Robbers. We investigate the cheating robot number for planar graphs and give a\u0000tight bound for bipartite planar graphs. We show that determining whether a\u0000graph has a cheating robot number at most fixed $k$ can be done in polynomial\u0000time. We also obtain bounds on the cheating robot number for strong and\u0000lexicographic products of graphs.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255042","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}
The monography considers the problem of constructing a Hamiltonian cycle in a complete graph. A rule for constructing a Hamiltonian cycle based on isometric cycles of a graph is established. An algorithm for constructing a Hamiltonian cycle based on ring summation of isometric cycles of a graph is presented. Based on the matrix of distances between vertices, the weight of each cycle is determined as an additive sum of the weights of its edges. To construct an optimal route of a graph, the basic idea of finding an optimal route between four vertices is used. Further successive constructions are aimed at joining an adjacent isometric cycle with an increase in the number of vertices by one unit. The recursive process continues until all vertices of the graph are connected. Based on the introduced mathematical apparatus, the monography presents a new algorithm for solving the symmetric Traveling salesman problem. Some examples of solving the problem are provided.
{"title":"Algorithmic methods of finite discrete structures. Hamiltonian cycle of a complete graph and the Traveling salesman problem","authors":"Sergey Kurapov, Maxim Davidovsky, Svetlana Polyuga","doi":"arxiv-2409.11563","DOIUrl":"https://doi.org/arxiv-2409.11563","url":null,"abstract":"The monography considers the problem of constructing a Hamiltonian cycle in a\u0000complete graph. A rule for constructing a Hamiltonian cycle based on isometric\u0000cycles of a graph is established. An algorithm for constructing a Hamiltonian\u0000cycle based on ring summation of isometric cycles of a graph is presented.\u0000Based on the matrix of distances between vertices, the weight of each cycle is\u0000determined as an additive sum of the weights of its edges. To construct an\u0000optimal route of a graph, the basic idea of finding an optimal route between\u0000four vertices is used. Further successive constructions are aimed at joining an\u0000adjacent isometric cycle with an increase in the number of vertices by one\u0000unit. The recursive process continues until all vertices of the graph are\u0000connected. Based on the introduced mathematical apparatus, the monography\u0000presents a new algorithm for solving the symmetric Traveling salesman problem.\u0000Some examples of solving the problem are provided.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255057","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}
The $lambda$-backbone coloring of the graph $G$ with backbone $H$ is a graph-coloring problem in which we are given a graph $G$ and a subgraph $H$, and we want to assign colors to vertices in such a way that the endpoints of every edge from $G$ have different colors, and the endpoints of every edge from $H$ are assigned colors which differ by at least $lambda$. In this paper we pursue research on backbone coloring of bounded-degree graphs with well-known classes of backbones. Our result is an almost complete classification of problems in the form $BBC_{lambda}(G, H) le lambda + k$ for graphs with maximum degree $4$ and backbones from the following classes: paths, trees, matchings, and galaxies.
{"title":"Backbone coloring for graphs with degree 4","authors":"Krzysztof Michalik, Krzysztof Turowski","doi":"arxiv-2409.10201","DOIUrl":"https://doi.org/arxiv-2409.10201","url":null,"abstract":"The $lambda$-backbone coloring of the graph $G$ with backbone $H$ is a\u0000graph-coloring problem in which we are given a graph $G$ and a subgraph $H$,\u0000and we want to assign colors to vertices in such a way that the endpoints of\u0000every edge from $G$ have different colors, and the endpoints of every edge from\u0000$H$ are assigned colors which differ by at least $lambda$. In this paper we pursue research on backbone coloring of bounded-degree\u0000graphs with well-known classes of backbones. Our result is an almost complete\u0000classification of problems in the form $BBC_{lambda}(G, H) le lambda + k$\u0000for graphs with maximum degree $4$ and backbones from the following classes:\u0000paths, trees, matchings, and galaxies.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"104 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269003","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}
In this paper, we determine the maximum size of a nonhamiltonian-connected graph with prescribed order and minimum degree. We also characterize the extremal graphs that attain this maximum size. This work generalizes a previous result obtained by Ore [ J. Math. Pures Appl. 42 (1963) 21-27] and further extends a theorem proved by Ho, Lin, Tan, Hsu, and Hsu [Appl. Math. Lett. 23 (2010) 26-29]. As a corollary of our main result, we determine the maximum size of a $k$-connected nonhamiltonian-connected graph with a given order.
在本文中，我们确定了具有规定阶数和最小度数的非哈密顿连接图的最大尺寸。我们还描述了达到这个最大尺寸的极端图的特征。这项工作概括了之前由 Ore [ J. Math. Pures Appl.作为我们主要结果的推论，我们确定了具有给定阶的 $k$ 连接非哈密顿连接图的最大尺寸。
{"title":"The maximum size of a nonhamiltonian-connected graph with given order and minimum degree","authors":"Leilei Zhang","doi":"arxiv-2409.10255","DOIUrl":"https://doi.org/arxiv-2409.10255","url":null,"abstract":"In this paper, we determine the maximum size of a nonhamiltonian-connected\u0000graph with prescribed order and minimum degree. We also characterize the\u0000extremal graphs that attain this maximum size. This work generalizes a previous\u0000result obtained by Ore [ J. Math. Pures Appl. 42 (1963) 21-27] and further\u0000extends a theorem proved by Ho, Lin, Tan, Hsu, and Hsu [Appl. Math. Lett. 23\u0000(2010) 26-29]. As a corollary of our main result, we determine the maximum size\u0000of a $k$-connected nonhamiltonian-connected graph with a given order.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269002","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}
A graph is chordal if it does not contain an induced cycle of length greater than three. We determine the minimum size of a chordal graph with given order and minimum degree. In doing so, we have discovered interesting properties of chordal graphs.
{"title":"The minimum size of a chordal graph with given order and minimum degree","authors":"Xingzhi Zhan, Leilei Zhang","doi":"arxiv-2409.10261","DOIUrl":"https://doi.org/arxiv-2409.10261","url":null,"abstract":"A graph is chordal if it does not contain an induced cycle of length greater\u0000than three. We determine the minimum size of a chordal graph with given order\u0000and minimum degree. In doing so, we have discovered interesting properties of\u0000chordal graphs.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255069","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}
noindent A perfect dominating set in a graph $G=(V,E)$ is a subset $S subseteq V$ such that each vertex in $V setminus S$ has exactly one neighbor in $S$. A perfect coalition in $G$ consists of two disjoint sets of vertices $V_i$ and $V_j$ such that i) neither $V_i$ nor $V_j$ is a dominating set, ii) each vertex in $V(G) setminus V_i$ has at most one neighbor in $V_i$ and each vertex in $V(G) setminus V_j$ has at most one neighbor in $V_j$, and iii) $V_i cup V_j$ is a perfect dominating set. A perfect coalition partition (abbreviated $prc$-partition) in a graph $G$ is a vertex partition $pi= lbrace V_1,V_2,dots ,V_k rbrace$ such that for each set $V_i$ of $pi$ either $V_i$ is a singleton dominating set, or there exists a set $V_j in pi$ that forms a perfect coalition with $V_i$. In this paper, we initiate the study of perfect coalition partitions in graphs. We obtain a bound on the number of perfect coalitions involving each member of a perfect coalition partition, in terms of maximum degree. The perfect coalition of some special graphs are investigated. The graph $G$ with $delta(G)=1$, the triangle-free graphs $G$ with prefect coalition number of order of $G$ and the trees $T$ with prefect coalition number in ${n,n-1,n-2}$ where $n=|V(T)|$ are characterized.
{"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}
The existence of $srg(99,14,1,2)$ has been a question of interest for several decades to the moment. In this paper we consider the structural properties in general for the family of strongly regular graphs with parameters $lambda =1$ and $mu =2$. In particular, we establish the lower bound for the number of hexagons and, by doing that, we show the connection between the existence of the aforementioned graph and the number of its hexagons.
{"title":"The Lower Bound for Number of Hexagons in Strongly Regular Graphs with Parameters $λ=1$ and $μ=2$","authors":"Reimbay Reimbayev","doi":"arxiv-2409.10620","DOIUrl":"https://doi.org/arxiv-2409.10620","url":null,"abstract":"The existence of $srg(99,14,1,2)$ has been a question of interest for several\u0000decades to the moment. In this paper we consider the structural properties in\u0000general for the family of strongly regular graphs with parameters $lambda =1$\u0000and $mu =2$. In particular, we establish the lower bound for the number of\u0000hexagons and, by doing that, we show the connection between the existence of\u0000the aforementioned graph and the number of its hexagons.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268999","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}
In 1960, Nash-Williams proved his strong orientation theorem that every finite graph has an orientation in which the number of directed paths between any two vertices is at least half the number of undirected paths between them (rounded down). Nash-Williams conjectured that it is possible to find such orientations for infinite graphs as well. We provide a partial answer by proving that all rayless graphs have such an orientation.
{"title":"The strong Nash-Williams orientation theorem for rayless graphs","authors":"Max Pitz, Jacob Stegemann","doi":"arxiv-2409.10378","DOIUrl":"https://doi.org/arxiv-2409.10378","url":null,"abstract":"In 1960, Nash-Williams proved his strong orientation theorem that every\u0000finite graph has an orientation in which the number of directed paths between\u0000any two vertices is at least half the number of undirected paths between them\u0000(rounded down). Nash-Williams conjectured that it is possible to find such\u0000orientations for infinite graphs as well. We provide a partial answer by\u0000proving that all rayless graphs have such an orientation.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255068","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}
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