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Spanning trees in graphs without large bipartite holes 无大二部孔图中的生成树
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-11-14 DOI: 10.1017/s0963548323000378
Jie Han, Jie Hu, Lidan Ping, Guanghui Wang, Yi Wang, Donglei Yang
{"title":"Spanning trees in graphs without large bipartite holes","authors":"Jie Han, Jie Hu, Lidan Ping, Guanghui Wang, Yi Wang, Donglei Yang","doi":"10.1017/s0963548323000378","DOIUrl":"https://doi.org/10.1017/s0963548323000378","url":null,"abstract":"Abstract We show that for any $varepsilon gt 0$ and $Delta in mathbb{N}$ , there exists $alpha gt 0$ such that for sufficiently large $n$ , every $n$ -vertex graph $G$ satisfying that $delta (G)geq varepsilon n$ and $e(X, Y)gt 0$ for every pair of disjoint vertex sets $X, Ysubseteq V(G)$ of size $alpha n$ contains all spanning trees with maximum degree at most $Delta$ . This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134954546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Approximate discrete entropy monotonicity for log-concave sums 对数凹和的近似离散熵单调性
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-11-13 DOI: 10.1017/s0963548323000408
Lampros Gavalakis
{"title":"Approximate discrete entropy monotonicity for log-concave sums","authors":"Lampros Gavalakis","doi":"10.1017/s0963548323000408","DOIUrl":"https://doi.org/10.1017/s0963548323000408","url":null,"abstract":"Abstract It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n geq 1$ , if $X_1,ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then begin{equation*} H(X_1+cdots +X_{n+1}) geq H(X_1+cdots +X_{n}) + frac {1}{2}log {Bigl (frac {n+1}{n}Bigr )} - o(1) end{equation*} as $H(X_1) to infty$ , where $H(X_1)$ denotes the (discrete) Shannon entropy. The problem is reduced to the continuous setting by showing that if $U_1,ldots,U_n$ are independent continuous uniforms on $(0,1)$ , then begin{equation*} h(X_1+cdots +X_n + U_1+cdots +U_n) = H(X_1+cdots +X_n) + o(1), end{equation*} as $H(X_1) to infty$ , where $h$ stands for the differential entropy. Explicit bounds for the $o(1)$ -terms are provided.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136347182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 special case of Vu’s conjecture: colouring nearly disjoint graphs of bounded maximum degree Vu猜想的一个特例:最大有界度的近不相交图的上色
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-11-10 DOI: 10.1017/s0963548323000299
Tom Kelly, Daniela Kühn, Deryk Osthus
{"title":"A special case of Vu’s conjecture: colouring nearly disjoint graphs of bounded maximum degree","authors":"Tom Kelly, Daniela Kühn, Deryk Osthus","doi":"10.1017/s0963548323000299","DOIUrl":"https://doi.org/10.1017/s0963548323000299","url":null,"abstract":"Abstract A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$ , then the following holds. For every fixed $C$ , if each vertex $v in bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, dots, G_m$ , then the (list) chromatic number of $bigcup _{i=1}^m G_i$ is at most $D + o(D)$ . This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135092240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 1
On oriented cycles in randomly perturbed digraphs 随机摄动有向图中的有向环
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-11-08 DOI: 10.1017/s0963548323000391
Igor Araujo, József Balogh, Robert A. Krueger, Simón Piga, Andrew Treglown
{"title":"On oriented cycles in randomly perturbed digraphs","authors":"Igor Araujo, József Balogh, Robert A. Krueger, Simón Piga, Andrew Treglown","doi":"10.1017/s0963548323000391","DOIUrl":"https://doi.org/10.1017/s0963548323000391","url":null,"abstract":"Abstract In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $alpha gt 0$ , there exists a constant $C$ such that for every $n$ -vertex digraph of minimum semi-degree at least $alpha n$ , if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$ . Our proofs make use of a variant of an absorbing method of Montgomery.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 4
Mastermind with a linear number of queries 具有线性查询数的策划者
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-11-08 DOI: 10.1017/s0963548323000366
Anders Martinsson, Pascal Su
{"title":"Mastermind with a linear number of queries","authors":"Anders Martinsson, Pascal Su","doi":"10.1017/s0963548323000366","DOIUrl":"https://doi.org/10.1017/s0963548323000366","url":null,"abstract":"Abstract Since the 1960s Mastermind has been studied for the combinatorial and information-theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colours. For $k$ colours and $n$ positions, Chvátal found asymptotically optimal bounds when $k le n^{1-varepsilon }$ . Following a sequence of gradual improvements for $kgeq n$ colours, the central open question is to resolve the gap between $Omega (n)$ and $mathcal{O}(nlog log n)$ for $k=n$ . In this paper, we resolve this gap by presenting the first algorithm for solving $k=n$ Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters $k$ and $n$ .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 6
On the choosability of -minor-free graphs 无次图的可选择性
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-11-03 DOI: 10.1017/s0963548323000354
Olivier Fischer, Raphael Steiner
{"title":"On the choosability of -minor-free graphs","authors":"Olivier Fischer, Raphael Steiner","doi":"10.1017/s0963548323000354","DOIUrl":"https://doi.org/10.1017/s0963548323000354","url":null,"abstract":"Abstract Given a graph $H$ , let us denote by $f_chi (H)$ and $f_ell (H)$ , respectively, the maximum chromatic number and the maximum list chromatic number of $H$ -minor-free graphs. Hadwiger’s famous colouring conjecture from 1943 states that $f_chi (K_t)=t-1$ for every $t ge 2$ . A closely related problem that has received significant attention in the past concerns $f_ell (K_t)$ , for which it is known that $2t-o(t) le f_ell (K_t) le O(t (!log log t)^6)$ . Thus, $f_ell (K_t)$ is bounded away from the conjectured value $t-1$ for $f_chi (K_t)$ by at least a constant factor. The so-called $H$ -Hadwiger’s conjecture, proposed by Seymour, asks to prove that $f_chi (H)={textrm{v}}(H)-1$ for a given graph $H$ (which would be implied by Hadwiger’s conjecture). In this paper, we prove several new lower bounds on $f_ell (H)$ , thus exploring the limits of a list colouring extension of $H$ -Hadwiger’s conjecture. Our main results are: For every $varepsilon gt 0$ and all sufficiently large graphs $H$ we have $f_ell (H)ge (1-varepsilon )({textrm{v}}(H)+kappa (H))$ , where $kappa (H)$ denotes the vertex-connectivity of $H$ . For every $varepsilon gt 0$ there exists $C=C(varepsilon )gt 0$ such that asymptotically almost every $n$ -vertex graph $H$ with $left lceil C nlog nright rceil$ edges satisfies $f_ell (H)ge (2-varepsilon )n$ . The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of $H$ -minor-free graphs is separated from the desired value of $({textrm{v}}(H)-1)$ by a constant factor for all large graphs $H$ of linear connectivity. The second result tells us that for almost all graphs $H$ with superlogarithmic average degree $f_ell (H)$ is separated from $({textrm{v}}(H)-1)$ by a constant factor arbitrarily close to $2$ . Conceptually these results indicate that the graphs $H$ for which $f_ell (H)$ is close to the conjectured value $({textrm{v}}(H)-1)$ for $f_chi (H)$ are typically rather sparse.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135868821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Spanning subdivisions in Dirac graphs 在狄拉克图中生成细分
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-10-20 DOI: 10.1017/s0963548323000342
Matías Pavez-Signé
{"title":"Spanning subdivisions in Dirac graphs","authors":"Matías Pavez-Signé","doi":"10.1017/s0963548323000342","DOIUrl":"https://doi.org/10.1017/s0963548323000342","url":null,"abstract":"Abstract We show that for every $nin mathbb N$ and $log nle dlt n$ , if a graph $G$ has $N=Theta (dn)$ vertices and minimum degree $(1+o(1))frac{N}{2}$ , then it contains a spanning subdivision of every $n$ -vertex $d$ -regular graph.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135570152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 1
Many Hamiltonian subsets in large graphs with given density 给定密度的大图中的许多哈密顿子集
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-10-02 DOI: 10.1017/s0963548323000317
Stijn Cambie, Jun Gao, Hong Liu
{"title":"Many Hamiltonian subsets in large graphs with given density","authors":"Stijn Cambie, Jun Gao, Hong Liu","doi":"10.1017/s0963548323000317","DOIUrl":"https://doi.org/10.1017/s0963548323000317","url":null,"abstract":"Abstract A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh, and Staden proved that for large $d$ , among all graphs with minimum degree $d$ , $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ( $approx 2^{d+1}$ ). Among others, our bound implies that an $n$ -vertex $C_4$ -free graph with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Intersecting families without unique shadow 交叉的家族没有独特的影子
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-10-02 DOI: 10.1017/s0963548323000305
Peter Frankl, Jian Wang
{"title":"Intersecting families without unique shadow","authors":"Peter Frankl, Jian Wang","doi":"10.1017/s0963548323000305","DOIUrl":"https://doi.org/10.1017/s0963548323000305","url":null,"abstract":"Abstract Let $mathcal{F}$ be an intersecting family. A $(k-1)$ -set $E$ is called a unique shadow if it is contained in exactly one member of $mathcal{F}$ . Let ${mathcal{A}}={Ain binom{[n]}{k}colon |Acap {1,2,3}|geq 2}$ . In the present paper, we show that for $ngeq 28k$ , $mathcal{A}$ is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other results of a similar flavour are established as well.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135893529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 1
The Excluded Tree Minor Theorem Revisited 重新考察排除树小定理
4区 数学
Combinatorics, Probability & Computing Pub Date : 2023-09-27 DOI: 10.1017/s0963548323000275
Vida Dujmović, Robert Hickingbotham, Gwenaël Joret, Piotr Micek, Pat Morin, David R. Wood
{"title":"The Excluded Tree Minor Theorem Revisited","authors":"Vida Dujmović, Robert Hickingbotham, Gwenaël Joret, Piotr Micek, Pat Morin, David R. Wood","doi":"10.1017/s0963548323000275","DOIUrl":"https://doi.org/10.1017/s0963548323000275","url":null,"abstract":"Abstract We prove that for every tree $T$ of radius $h$ , there is an integer $c$ such that every $T$ -minor-free graph is contained in $Hboxtimes K_c$ for some graph $H$ with pathwidth at most $2h-1$ . This is a qualitative strengthening of the Excluded Tree Minor Theorem of Robertson and Seymour (GM I). We show that radius is the right parameter to consider in this setting, and $2h-1$ is the best possible bound.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135536654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
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