arXiv - MATH - Combinatorics最新文献_第10页

Induced subgraphs of $K_r$-free graphs and the Erdős--Rogers problem 无 K_r$ 图的诱导子图和厄尔多斯--罗杰斯问题

arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06650

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":"arxiv-2409.06650","DOIUrl":"https://doi.org/arxiv-2409.06650","url":null,"abstract":"For two graphs $F,H$ and a positive integer $n$, the function $f_{F,H}(n)$\u0000denotes the largest $m$ such that every $H$-free graph on $n$ vertices contains\u0000an $F$-free induced subgraph on $m$ vertices. This function has been\u0000extensively studied in the last 60 years when $F$ and $H$ are cliques and\u0000became known as the ErdH{o}s-Rogers function. Recently, Balogh, Chen and Luo,\u0000and Mubayi and Verstra\"ete initiated the systematic study of this function in\u0000the case where $F$ is a general graph. Answering, in a strong form, a question of Mubayi and Verstra\"ete, we prove\u0000that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there\u0000exists some $varepsilon_F>0$ such that\u0000$f_{F,K_r}(n)=O(n^{1/2-varepsilon_F})$. This result is tight in two ways.\u0000Firstly, it is no longer true if $F$ contains $K_{r-1}$ as a subgraph.\u0000Secondly, we show that for all $rgeq 4$ and $varepsilon>0$, there exists a\u0000$K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=Omega(n^{1/2-varepsilon})$.\u0000Along the way of proving this, we show in particular that for every graph $F$\u0000with minimum degree $t$, we have $f_{F,K_4}(n)=Omega(n^{1/2-6/sqrt{t}})$.\u0000This answers (in a strong form) another question of Mubayi and Verstra\"ete.\u0000Finally, we prove that there exist absolute constants $0<c<C$ such that for\u0000each $rgeq 4$, if $F$ is a bipartite graph with sufficiently large minimum\u0000degree, then $Omega(n^{frac{c}{log r}})leq f_{F,K_r}(n)leq\u0000O(n^{frac{C}{log r}})$. This shows that for graphs $F$ with large minimum\u0000degree, the behaviour of $f_{F,K_r}(n)$ is drastically different from that of\u0000the corresponding off-diagonal Ramsey number $f_{K_2,K_r}(n)$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197627","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

Analogues of Bermond-Bollobás Conjecture for Cages Yield Expander Families 贝蒙-波洛巴猜想的类比笼产生扩张器家族

arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06629

Leonard Chidiebere Eze, Robert Jajcay

引用次数: 0

Random embeddings of bounded degree trees with optimal spread 有界度树的随机嵌入与最佳扩散

arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06640

Paul Bastide, Clément Legrand-Duchesne, Alp Müyesser

{"title":"Random embeddings of bounded degree trees with optimal spread","authors":"Paul Bastide, Clément Legrand-Duchesne, Alp Müyesser","doi":"arxiv-2409.06640","DOIUrl":"https://doi.org/arxiv-2409.06640","url":null,"abstract":"A seminal result of Koml'os, S'ark\"ozy, and Szemer'edi states that any\u0000n-vertex graph G with minimum degree at least (1/2 + {alpha})n contains every\u0000n-vertex tree T of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin\u0000extended this result to show that such graphs G in fact support an optimally\u0000spread distribution on copies of a given T, which implies, using the recent\u0000breakthroughs on the Kahn-Kalai conjecture, the robustness result that T is a\u0000subgraph of sparse random subgraphs of G as well. Pham, Sah, Sawhney, and\u0000Simkin construct their optimally spread distribution by following closely the\u0000original proof of the Koml'os-S'ark\"ozy-Szemer'edi theorem which uses the\u0000blow-up lemma and the Szemer'edi regularity lemma. We give an alternative,\u0000regularity-free construction that instead uses the\u0000Koml'os-S'ark\"ozy-Szemer'edi theorem (which has a regularity-free proof due\u0000to Kathapurkar and Montgomery) as a black-box. Our proof is based on the simple\u0000and general insight that, if G has linear minimum degree, almost all constant\u0000sized subgraphs of G inherit the same minimum degree condition that G has.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197628","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 structure of Hurwitz numbers with fixed ramification profile and varying genus 具有固定斜切剖面和不同属数的赫维兹数的结构

arXiv - MATH - Combinatorics Pub Date : 2024-09-10 DOI: arxiv-2409.06655

Norman Do, Jian He, Heath Robertson

{"title":"The structure of Hurwitz numbers with fixed ramification profile and varying genus","authors":"Norman Do, Jian He, Heath Robertson","doi":"arxiv-2409.06655","DOIUrl":"https://doi.org/arxiv-2409.06655","url":null,"abstract":"In 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$,\u0000branched covers of the Riemann sphere with simple ramification over prescribed\u0000points and no branching elsewhere. He showed that for fixed degree $d$, the\u0000enumeration possesses a remarkable structure. More precisely, it can be\u0000expressed as a linear combination of exponentials $m^{2g-2+2d}$, where $m$\u0000ranges over the integers from $1$ to $binom{d}{2}$. In this paper, we generalise this structural result to Hurwitz numbers that\u0000enumerate branched covers which also have a prescribed ramification profile\u0000over one point. Our proof fundamentally uses the infinite wedge space, in\u0000particular the connected correlators of products of $mathcal{E}$-operators.\u0000The recent study of Hurwitz numbers has often focussed on their structure with\u0000fixed genus and varying ramification profile. Our main result is orthogonal to\u0000this, allowing for the explicit calculation and the asymptotic analysis of\u0000Hurwitz numbers in large genus. We pose the broad question of which other enumerative problems exhibit\u0000analogous structure. We prove that orbifold Hurwitz numbers can also be\u0000expressed as a linear combination of exponentials and conjecture that monotone\u0000Hurwitz numbers share a similar structure, but with the inclusion of an\u0000additional linear term.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225150","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

Sharp Bounds for Generalized Zagreb Indices of Graphs 图的广义萨格勒布指数的锐界

arXiv - MATH - Combinatorics Pub Date : 2024-09-09 DOI: arxiv-2409.06081

Sanju Vaidya, Jeff Chang

引用次数: 0

Brunn-Minkowski type estimates for certain discrete sumsets 某些离散和集的布伦-闵科夫斯基类型估计

arXiv - MATH - Combinatorics Pub Date : 2024-09-09 DOI: arxiv-2409.05638

Albert Lopez Bruch, Yifan Jing, Akshat Mudgal

引用次数: 0

On the structure of extremal point-line arrangements 论极值点线排列的结构

arXiv - MATH - Combinatorics Pub Date : 2024-09-09 DOI: arxiv-2409.06115

Gabriel Currier, Jozsef Solymosi, Hung-Hsun Hans Yu