Combinatorica最新文献_第2页

Bounds on the Mod 2 Homology of Random 2-Dimensional Determinantal Hypertrees 随机二维行列式超树的模2同调的界

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-14 DOI: 10.1007/s00493-025-00142-6

András Mészáros

引用次数: 0

Supersaturation Beyond Color-Critical Graphs 超过颜色临界图的过饱和

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-14 DOI: 10.1007/s00493-025-00143-5

Jie Ma, Long-Tu Yuan

{"title":"Supersaturation Beyond Color-Critical Graphs","authors":"Jie Ma, Long-Tu Yuan","doi":"10.1007/s00493-025-00143-5","DOIUrl":"https://doi.org/10.1007/s00493-025-00143-5","url":null,"abstract":"The supersaturation problem for a given graph F asks for the minimum number (h_F(n,q)) of copies of F in an n-vertex graph with (textrm{ex}(n,F)+q) edges. Subsequent works by Rademacher, Erdős, and Lovász and Simonovits determine the optimal range of q (which is linear in n) for cliques F such that (h_F(n,q)) equals the minimum number (t_F(n,q)) of copies of F obtained from a maximum F-free n-vertex graph by adding q new edges. A breakthrough result of Mubayi extends this line of research from cliques to color-critical graphs F, and this was further strengthened by Pikhurko and Yilma who established the equality (h_F(n,q)=t_F(n,q)) for (1le qle epsilon _F n) and sufficiently large n. In this paper, we present several results on the supersaturation problem that extend beyond the existing framework. Firstly, we explicitly construct infinitely many graphs F with restricted properties for which (h_F(n,q)<qcdot t_F(n,1)) holds when (ngg qge 4), thus refuting a conjecture of Mubayi. Secondly, we extend the result of Pikhurko–Yilma by showing the equality (h_F(n,q)=t_F(n,q)) in the range (1le qle epsilon _F n) for any member F in a diverse and abundant graph family (which includes color-critical graphs, disjoint unions of cliques (K_r), and the Petersen graph). Lastly, we prove the existence of a graph F for any positive integer s such that (h_F(n,q)=t_F(n,q)) holds when (1le qle epsilon _F n^{1-1/s}), and (h_F(n,q)<t_F(n,q)) when (n^{1-1/s}/epsilon _Fle qle epsilon _F n), indicating that (q=Theta (n^{1-1/s})) serves as the threshold for the equality (h_F(n,q)=t_F(n,q)). We also discuss some additional remarks and related open problems.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"183 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618494","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

Gadget Construction and Structural Convergence 小工具构造与结构收敛

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-14 DOI: 10.1007/s00493-025-00140-8

David Hartman, Tomáš Hons, Jaroslav Nešetřil

引用次数: 0

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