Pub Date : 2023-11-14DOI: 10.1017/s0963548323000378
Jie Han, Jie Hu, Lidan Ping, Guanghui Wang, Yi Wang, Donglei Yang
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.
{"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}
Pub Date : 2023-11-13DOI: 10.1017/s0963548323000408
Lampros Gavalakis
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.
{"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}
Pub Date : 2023-11-10DOI: 10.1017/s0963548323000299
Tom Kelly, Daniela Kühn, Deryk Osthus
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.
{"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}
Pub Date : 2023-11-08DOI: 10.1017/s0963548323000391
Igor Araujo, József Balogh, Robert A. Krueger, Simón Piga, Andrew Treglown
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.
{"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}
Pub Date : 2023-11-08DOI: 10.1017/s0963548323000366
Anders Martinsson, Pascal Su
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 $n$ .
自20世纪60年代以来，人们一直在研究《智囊》游戏所提供的组合和信息理论兴趣。许多结果都是从Erdős和r nyi开始的，它们决定了两种颜色所需的最佳查询次数。对于$k$的颜色和$n$的位置，Chvátal找到渐近最优界，当$k le n^{1-varepsilon }$。随着对$kgeq n$颜色的一系列逐步改进，中心的开放问题是解决$k=n$的$Omega (n)$和$mathcal{O}(nlog log n)$之间的差距。在本文中，我们通过提出第一个用线性查询数求解$k=n$ Mastermind的算法来解决这一差距。因此，我们能够确定Mastermind对任何参数$k$和$n$的查询复杂度。
{"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}
Pub Date : 2023-11-03DOI: 10.1017/s0963548323000354
Olivier Fischer, Raphael Steiner
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.
{"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}
Pub Date : 2023-10-20DOI: 10.1017/s0963548323000342
Matías Pavez-Signé
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.
{"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}
Pub Date : 2023-10-02DOI: 10.1017/s0963548323000317
Stijn Cambie, Jun Gao, Hong Liu
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.
一个图中的顶点集合如果能引出一个包含哈密顿循环的子图，则称为哈密顿子集。Kim, Liu, Sharifzadeh, and Staden证明了对于大$d$，在所有具有最小度$d$的图中，$K_{d+1}$使哈密顿子集的个数最小。我们证明了一个考虑图的顺序和结构的近似最优下界。对于许多自然图类，它提供了一个比极值界($approx 2^{d+1}$)更好的界。其中，我们的界表明一个$n$顶点$C_4$最小度$d$的无图包含至少$n2^{d^{2- 0(1)}}$哈密顿子集。
{"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}
Pub Date : 2023-10-02DOI: 10.1017/s0963548323000305
Peter Frankl, Jian Wang
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.
{"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}
Pub Date : 2023-09-27DOI: 10.1017/s0963548323000275
Vida Dujmović, Robert Hickingbotham, Gwenaël Joret, Piotr Micek, Pat Morin, David R. Wood
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.
{"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}
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