Combinatorics, Probability and Computing最新文献

On the Ramsey numbers of daisies II 关于雏菊的拉姆齐数 II

Combinatorics, Probability and Computing Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000208

Marcelo Sales

引用次数: 0

List packing number of bounded degree graphs 有界阶数图的包装数列表

Combinatorics, Probability and Computing Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000191

Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang

{"title":"List packing number of bounded degree graphs","authors":"Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang","doi":"10.1017/s0963548324000191","DOIUrl":"https://doi.org/10.1017/s0963548324000191","url":null,"abstract":"We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $Delta$ has list packing number at most $Delta +1$ . Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks’-type theorem for the list packing number.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267798","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

A new formula for the determinant and bounds on its tensor and Waring ranks 行列式的新公式及其张量和瓦林等级的界限

Combinatorics, Probability and Computing Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000233

Robin Houston, Adam P. Goucher, Nathaniel Johnston

{"title":"A new formula for the determinant and bounds on its tensor and Waring ranks","authors":"Robin Houston, Adam P. Goucher, Nathaniel Johnston","doi":"10.1017/s0963548324000233","DOIUrl":"https://doi.org/10.1017/s0963548324000233","url":null,"abstract":"We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n times n$ determinant tensor is no larger than the $n$ -th Bell number, which is much smaller than the previously best-known upper bounds when $n geq 4$ . Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 times 4$ determinant over ${mathbb{F}}_2$ has tensor rank exactly equal to $12$ . Our results also improve upon the best-known upper bound for the Waring rank of the determinant when $n geq 17$ , and lead to a new family of axis-aligned polytopes that tile ${mathbb{R}}^n$ .","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267795","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

On the Ramsey numbers of daisies I 关于拉姆齐雏菊的数量 I

Combinatorics, Probability and Computing Pub Date : 2024-09-18 DOI: 10.1017/s0963548324000221

Pavel Pudlák, Vojtech Rödl, Marcelo Sales

{"title":"On the Ramsey numbers of daisies I","authors":"Pavel Pudlák, Vojtech Rödl, Marcelo Sales","doi":"10.1017/s0963548324000221","DOIUrl":"https://doi.org/10.1017/s0963548324000221","url":null,"abstract":"Daisies are a special type of hypergraph introduced by Bollobás, Leader and Malvenuto. An $r$ -daisy determined by a pair of disjoint sets $K$ and $M$ is the $(r+|K|)$ -uniform hypergraph ${Kcup P,{:}, Pin M^{(r)}}$ . Bollobás, Leader and Malvenuto initiated the study of Turán type density problems for daisies. This paper deals with Ramsey numbers of daisies, which are natural generalisations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of $r$ -daisies and also for special cases where the size of the kernel is bounded.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267796","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

Counting spanning subgraphs in dense hypergraphs 计算密集超图中的跨越子图

Combinatorics, Probability and Computing Pub Date : 2024-05-30 DOI: 10.1017/s0963548324000178

Richard Montgomery, Matías Pavez-Signé

{"title":"Counting spanning subgraphs in dense hypergraphs","authors":"Richard Montgomery, Matías Pavez-Signé","doi":"10.1017/s0963548324000178","DOIUrl":"https://doi.org/10.1017/s0963548324000178","url":null,"abstract":"We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $kgeq 2$ and $1leq ell leq k-1$ , we show that every $k$ -graph on $n$ vertices with minimum codegree at least begin{equation*} left {begin {array}{l@{quad}l} left (dfrac {1}{2}+o(1)right )n & text { if }(k-ell )mid k,[5pt] left (dfrac {1}{lceil frac {k}{k-ell }rceil (k-ell )}+o(1)right )n & text { if }(k-ell )nmid k, end {array} right . end{equation*} contains $exp!(nlog n-Theta (n))$ Hamilton $ell$ -cycles as long as $(k-ell )mid n$ . When $(k-ell )mid k$ , this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when <jats:al","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141188697","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

A generalisation of Varnavides’s theorem 瓦纳维德斯定理的一般化

Combinatorics, Probability and Computing Pub Date : 2024-05-29 DOI: 10.1017/s096354832400018x

Asaf Shapira

引用次数: 0

Noise sensitivity of the minimum spanning tree of the complete graph 完整图最小生成树的噪声敏感度

Combinatorics, Probability and Computing Pub Date : 2024-05-23 DOI: 10.1017/s0963548324000129

Omer Israeli, Yuval Peled

{"title":"Noise sensitivity of the minimum spanning tree of the complete graph","authors":"Omer Israeli, Yuval Peled","doi":"10.1017/s0963548324000129","DOIUrl":"https://doi.org/10.1017/s0963548324000129","url":null,"abstract":"We study the noise sensitivity of the minimum spanning tree (MST) of the $n$ -vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by $n^{1/3}$ and vertices are given a uniform measure, the MST converges in distribution in the Gromov–Hausdorff–Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability $varepsilon gg n^{-1/3}$ , then the pair of rescaled minimum spanning trees – before and after the noise – converges in distribution to independent random spaces. Conversely, if $varepsilon ll n^{-1/3}$ , the GHP distance between the rescaled trees goes to $0$ in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of $n^{-1/3}$ coincides with the critical window of the Erdős-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148786","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

Rainbow Hamiltonicity in uniformly coloured perturbed digraphs 均匀彩色扰动数图中的彩虹汉密尔顿性

Combinatorics, Probability and Computing Pub Date : 2024-05-13 DOI: 10.1017/s0963548324000130

Kyriakos Katsamaktsis, Shoham Letzter, Amedeo Sgueglia

{"title":"Rainbow Hamiltonicity in uniformly coloured perturbed digraphs","authors":"Kyriakos Katsamaktsis, Shoham Letzter, Amedeo Sgueglia","doi":"10.1017/s0963548324000130","DOIUrl":"https://doi.org/10.1017/s0963548324000130","url":null,"abstract":"We investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every $delta in (0,1)$ there exists $C = C(delta ) gt 0$ such that the following holds. Let $D_0$ be an $n$ -vertex digraph with minimum semidegree at least $delta n$ and suppose that each edge of the union of $D_0$ with a copy of the random digraph $mathbf{D}(n,C/n)$ on the same vertex set gets a colour in $[n]$ independently and uniformly at random. Then, with high probability, $D_0 cup mathbf{D}(n,C/n)$ has a rainbow directed Hamilton cycle. This improves a result of Aigner-Horev and Hefetz ((2021) SIAM J. Discrete Math.35(3) 1569–1577), who proved the same in the undirected setting when the edges are coloured","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930093","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

Maximum chordal subgraphs of random graphs 随机图的最大和弦子图

Combinatorics, Probability and Computing Pub Date : 2024-05-03 DOI: 10.1017/s0963548324000154

Michael Krivelevich, Maksim Zhukovskii

引用次数: 0

Turán problems in pseudorandom graphs 伪随机图中的图兰问题

Combinatorics, Probability and Computing Pub Date : 2024-04-29 DOI: 10.1017/s0963548324000142

Xizhi Liu, Dhruv Mubayi, David Munhá Correia

{"title":"Turán problems in pseudorandom graphs","authors":"Xizhi Liu, Dhruv Mubayi, David Munhá Correia","doi":"10.1017/s0963548324000142","DOIUrl":"https://doi.org/10.1017/s0963548324000142","url":null,"abstract":"Given a graph $F$ , we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$ . We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$ . Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$ -free graphs that are also triangle-free. Finally, we give a different proof for the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer, and Pepe that they have no large clique.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140812125","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