Combinatorica最新文献_第2页

Induced Subgraphs of $$K_r$$ -Free Graphs and the Erdős–Rogers Problem $$K_r$$自由图的诱导子图与Erdős-Rogers问题

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-27 DOI: 10.1007/s00493-025-00147-1

Lior Gishboliner, Oliver Janzer, Benny Sudakov

{"title":"Induced Subgraphs of $$K_r$$ -Free Graphs and the Erdős–Rogers Problem","authors":"Lior Gishboliner, Oliver Janzer, Benny Sudakov","doi":"10.1007/s00493-025-00147-1","DOIUrl":"https://doi.org/10.1007/s00493-025-00147-1","url":null,"abstract":"For two graphs F, H and a positive integer n, the function (f_{F,H}(n)) denotes the largest m such that every H-free graph on n vertices contains an F-free induced subgraph on m vertices. This function has been extensively studied in the last 60 years when F and H are cliques and became known as the Erdős–Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstraëte initiated the systematic study of this function in the case where F is a general graph. Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer r and every (K_{r-1})-free graph F, there exists some (varepsilon _F>0) such that (f_{F,K_r}(n)=O(n^{1/2-varepsilon _F})). This result is tight in two ways. Firstly, it is no longer true if F contains (K_{r-1}) as a subgraph. Secondly, we show that for all (rge 4) and (varepsilon >0), there exists a (K_{r-1})-free graph F for which (f_{F,K_r}(n)=Omega (n^{1/2-varepsilon })). Along the way of proving this, we show in particular that for every graph F with minimum degree t, we have (f_{F,K_4}(n)=Omega (n^{1/2-6/sqrt{t}})). This answers (in a strong form) another question of Mubayi and Verstraëte. Finally, we prove that there exist absolute constants (0<c<C) such that for each (rge 4), if F is a bipartite graph with sufficiently large minimum degree, then (Omega (n^{frac{c}{log r}})le f_{F,K_r}(n)le O(n^{frac{C}{log r}})). This shows that for graphs F with large minimum degree, the behaviour of (f_{F,K_r}(n)) is drastically different from that of the corresponding off-diagonal Ramsey number (f_{K_2,K_r}(n)).","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"57 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143713068","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 Large Family of Strongly Regular Graphs with Small Weisfeiler-Leman Dimension 一类具有小Weisfeiler-Leman维数的强正则图

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-24 DOI: 10.1007/s00493-025-00145-3

Jinzhuan Cai, Jin Guo, Alexander L. Gavrilyuk, Ilia Ponomarenko

引用次数: 0

The Signed Varchenko Determinant for Complexes of Oriented Matroids 有向拟阵复合体的有符号Varchenko行列式

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-17 DOI: 10.1007/s00493-025-00138-2

Winfried Hochstättler, Sophia Keip, Kolja Knauer

引用次数: 0

Improved Lower Bound Towards Chen–Chvátal Conjecture 改进的Chen-Chvátal猜想的下界

IF 1.1 2区数学

Combinatorica Pub Date : 2025-03-14 DOI: 10.1007/s00493-025-00137-3

Congkai Huang

引用次数: 0

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