Combinatorica最新文献_第4页

Sets of r-Graphs that Color All r-Graphs 为所有r-图着色的r-图集合

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-14 DOI: 10.1007/s00493-025-00144-4

Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf

{"title":"Sets of r-Graphs that Color All r-Graphs","authors":"Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf","doi":"10.1007/s00493-025-00144-4","DOIUrl":"https://doi.org/10.1007/s00493-025-00144-4","url":null,"abstract":"An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping (f:E(G) rightarrow E(H)) such that each r adjacent edges of G are mapped to r adjacent edges of H. For every (rge 3), let (mathcal H_r) be an inclusion-wise minimal set of connected r-graphs, such that for every connected r-graph G there is an (H in mathcal H_r) which colors G. The Petersen Coloring Conjecture states that (mathcal H_3) consists of the Petersen graph P. We show that if true, then this is a very exclusive situation. Our main result is that either (mathcal H_3 = {P}) or (mathcal H_3) is an infinite set and if (r ge 4), then (mathcal H_r) is an infinite set. In particular, for all (r ge 3), (mathcal H_r) is unique. We first characterize (mathcal H_r) and then prove that if (mathcal H_r) contains more than one element, then it is an infinite set. To obtain our main result we show that (mathcal H_r) contains the smallest r-graphs of class 2 and the smallest poorly matchable r-graphs, and we determine the smallest r-graphs of class 2.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"213 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Lower Bound for the Number of Pinned Angles Determined by a Cartesian Product Set 由笛卡尔积集确定的钉住角数的下界

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-07 DOI: 10.1007/s00493-025-00135-5

Oliver Roche-Newton

引用次数: 0

L-Systems and the Lovász Number l系统和Lovász数字

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-07 DOI: 10.1007/s00493-025-00136-4

William Linz

{"title":"L-Systems and the Lovász Number","authors":"William Linz","doi":"10.1007/s00493-025-00136-4","DOIUrl":"https://doi.org/10.1007/s00493-025-00136-4","url":null,"abstract":"Given integers (n> k > 0), and a set of integers (L subset [0, k-1]), an L-system is a family of sets (mathcal {F}subset left( {begin{array}{c}[n] kend{array}}right) ) such that (|F cap F'| in L) for distinct (F, F'in mathcal {F}). L-systems correspond to independent sets in a certain generalized Johnson graph G(n, k, L), so that the maximum size of an L-system is equivalent to finding the independence number of the graph G(n, k, L). The Lovász number (vartheta (G)) is a semidefinite programming approximation of the independence number (alpha ) of a graph G. In this paper, we determine the leading order term of (vartheta (G(n, k, L))) of any generalized Johnson graph with k and L fixed and (nrightarrow infty ). As an application of this theorem, we give an explicit construction of a graph G on n vertices with a large gap between the Lovász number and the Shannon capacity c(G). Specifically, we prove that for any (epsilon > 0), for infinitely many n there is a generalized Johnson graph G on n vertices which has ratio (vartheta (G)/c(G) = Omega (n^{1-epsilon })), which improves on all known constructions. The graph G a fortiori also has ratio (vartheta (G)/alpha (G) = Omega (n^{1-epsilon })), which greatly improves on the best known explicit construction.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"127 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143570294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Three-Cuts are a Charm: Acyclicity in 3-Connected Cubic Graphs 三切是一种魅力：三连通三次图的不环性

IF 1.1 2区数学

Combinatorica Pub Date : 2025-02-12 DOI: 10.1007/s00493-024-00126-y

František Kardoš, Edita Máčajová, Jean Paul Zerafa

{"title":"Three-Cuts are a Charm: Acyclicity in 3-Connected Cubic Graphs","authors":"František Kardoš, Edita Máčajová, Jean Paul Zerafa","doi":"10.1007/s00493-024-00126-y","DOIUrl":"https://doi.org/10.1007/s00493-024-00126-y","url":null,"abstract":"Let G be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the (S_4)-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of G such that the complement of their union is a bipartite subgraph of G. They actually show that given any (1^+)-factor F (a spanning subgraph of G such that its vertices have degree at least 1) and an arbitrary edge e of G, there exists a perfect matching M of G containing e such that (Gsetminus (Fcup M)) is bipartite. This is a step closer to comprehend better the Fan–Raspaud Conjecture and eventually the Berge–Fulkerson Conjecture. The (S_4)-Conjecture, now a theorem, is also the weakest assertion in a series of three conjectures made by Mazzuoccolo in 2013, with the next stronger statement being: there exist two perfect matchings of G such that the complement of their union is an acyclic subgraph of G. Unfortunately, this conjecture is not true: Jin, Steffen, and Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts. Here we show that, despite of this, every cyclically 3-edge-connected cubic graph satisfies this second conjecture.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"16 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143393285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

Constructing New Geometries: A Generalized Approach to Halving for Hypertopes 构建新几何图形：高位面减半的通用方法

IF 1.1 2区数学

Combinatorica Pub Date : 2025-01-16 DOI: 10.1007/s00493-024-00134-y

Claudio Alexandre Piedade, Philippe Tranchida

引用次数: 0

How Balanced Can Permutations Be? 排列如何平衡？

IF 1.1 2区数学

Combinatorica Pub Date : 2025-01-02 DOI: 10.1007/s00493-024-00127-x

Gal Beniamini, Nir Lavee, Nati Linial

引用次数: 0

The Number of Colorings of the Middle Layers of the Hamming Cube 汉明立方体中间层的着色数

IF 1.1 2区数学

Combinatorica Pub Date : 2025-01-02 DOI: 10.1007/s00493-024-00128-w

Lina Li, Gweneth McKinley, Jinyoung Park

引用次数: 0

Uniacute Spherical Codes 单锐角球码

IF 1.1 2区数学

Combinatorica Pub Date : 2025-01-02 DOI: 10.1007/s00493-024-00125-z

Saba Lepsveridze, Aleksandre Saatashvili, Yufei Zhao

{"title":"Uniacute Spherical Codes","authors":"Saba Lepsveridze, Aleksandre Saatashvili, Yufei Zhao","doi":"10.1007/s00493-024-00125-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00125-z","url":null,"abstract":"A spherical L-code, where (L subseteq [-1,infty )), consists of unit vectors in (mathbb {R}^d) whose pairwise inner products are contained in L. Determining the maximum cardinality (N_L(d)) of an L-code in (mathbb {R}^d) is a fundamental question in discrete geometry and has been extensively investigated for various choices of L. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to (L = {-alpha , alpha }), is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that (N_L(d) = O_L(d)) for (L = [-1, -beta ] cup {alpha }) with (alpha ,beta > 0) (we call such L-codes “uniacute”), leaving open the question of determining the leading constant factor. Balla, Dräxler, Keevash, and Sudakov proved a “uniform bound” showing (limsup _{drightarrow infty } N_L(d)/d le 2p) for (L = [-1, -beta ] cup {alpha }) and (p = lfloor alpha /beta rfloor + 1). For which ((alpha ,beta )) is this uniform bound tight? We completely answer this question. We develop a framework for studying uniacute codes, including a global structure theorem showing that the Gram matrix has an approximate p-block structure. We also formulate a notion of “modular codes,” which we conjecture to be optimal in high dimensions.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"375 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142916856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Excluded Minors for Embeddability into a Compact Surface 紧曲面可嵌入性的排除次元

IF 1.1 2区数学

Combinatorica Pub Date : 2024-12-29 DOI: 10.1007/s00493-024-00129-9

Agelos Georgakopoulos

引用次数: 0

Chiral Extensions of Regular Toroids 正则环面的手性扩展

IF 1.1 2区数学

Combinatorica Pub Date : 2024-12-29 DOI: 10.1007/s00493-024-00132-0

Antonio Montero, Micael Toledo

引用次数: 0