Pub Date : 2023-09-21DOI: 10.1017/s0963548323000330
David Galvin, Gwen McKinley, Will Perkins, Michail Sarantis, Prasad Tetali
Abstract We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $Delta$ have a zero-free disc almost as large as the optimal disc for graphs of maximum degree $Delta$ established by Shearer (of radius $sim 1/(e Delta )$ ). Up to logarithmic factors in $Delta$ this is optimal, even for hypergraphs with all edge sizes strictly greater than $2$ . We conjecture that for $kge 3$ , $k$ -uniform linear hypergraphs have a much larger zero-free disc of radius $Omega (Delta ^{- frac{1}{k-1}} )$ . We establish this in the case of linear hypertrees.
{"title":"On the zeroes of hypergraph independence polynomials","authors":"David Galvin, Gwen McKinley, Will Perkins, Michail Sarantis, Prasad Tetali","doi":"10.1017/s0963548323000330","DOIUrl":"https://doi.org/10.1017/s0963548323000330","url":null,"abstract":"Abstract We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $Delta$ have a zero-free disc almost as large as the optimal disc for graphs of maximum degree $Delta$ established by Shearer (of radius $sim 1/(e Delta )$ ). Up to logarithmic factors in $Delta$ this is optimal, even for hypergraphs with all edge sizes strictly greater than $2$ . We conjecture that for $kge 3$ , $k$ -uniform linear hypergraphs have a much larger zero-free disc of radius $Omega (Delta ^{- frac{1}{k-1}} )$ . We establish this in the case of linear hypertrees.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136235173","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 : 2023-09-18DOI: 10.1017/s0963548323000329
Piet Lammers, Fabio Toninelli
Abstract We study two models of discrete height functions , that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model , in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone , that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.
{"title":"Height function localisation on trees","authors":"Piet Lammers, Fabio Toninelli","doi":"10.1017/s0963548323000329","DOIUrl":"https://doi.org/10.1017/s0963548323000329","url":null,"abstract":"Abstract We study two models of discrete height functions , that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model , in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone , that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135154396","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 : 2023-06-26DOI: 10.1017/s0963548323000147
David Conlon, Rajko Nenadov, Miloš Trujić
Abstract We show that the size-Ramsey number of the $sqrt{n} times sqrt{n}$ grid graph is $O(n^{5/4})$ , improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.
{"title":"On the size-Ramsey number of grids","authors":"David Conlon, Rajko Nenadov, Miloš Trujić","doi":"10.1017/s0963548323000147","DOIUrl":"https://doi.org/10.1017/s0963548323000147","url":null,"abstract":"Abstract We show that the size-Ramsey number of the $sqrt{n} times sqrt{n}$ grid graph is $O(n^{5/4})$ , improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135608950","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 : 2023-06-13DOI: 10.1017/s0963548323000184
Simon Briend, Francisco Calvillo, Gábor Lugosi
Abstract We study the problem of finding the root vertex in large growing networks. We prove that it is possible to construct confidence sets of size independent of the number of vertices in the network that contain the root vertex with high probability in various models of random networks. The models include uniform random recursive dags and uniform Cooper-Frieze random graphs.
{"title":"Archaeology of random recursive dags and Cooper-Frieze random networks","authors":"Simon Briend, Francisco Calvillo, Gábor Lugosi","doi":"10.1017/s0963548323000184","DOIUrl":"https://doi.org/10.1017/s0963548323000184","url":null,"abstract":"Abstract We study the problem of finding the root vertex in large growing networks. We prove that it is possible to construct confidence sets of size independent of the number of vertices in the network that contain the root vertex with high probability in various models of random networks. The models include uniform random recursive dags and uniform Cooper-Frieze random graphs.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136101877","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 : 2023-06-09DOI: 10.1017/s0963548323000111
Dingding Dong, Nitya Mani, Yufei Zhao
Abstract We show that for a fixed $q$ , the number of $q$ -ary $t$ -error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t leq (1 - q^{-1})n - 2sqrt{n log n}$ , where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (log n)^{-2/3})$ .
{"title":"On the number of error correcting codes","authors":"Dingding Dong, Nitya Mani, Yufei Zhao","doi":"10.1017/s0963548323000111","DOIUrl":"https://doi.org/10.1017/s0963548323000111","url":null,"abstract":"Abstract We show that for a fixed $q$ , the number of $q$ -ary $t$ -error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t leq (1 - q^{-1})n - 2sqrt{n log n}$ , where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (log n)^{-2/3})$ .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135101085","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 : 2023-05-18DOI: 10.1017/s0963548323000123
Anurag Bishnoi, Simona Boyadzhiyska, Shagnik Das, Tamás Mészáros
Abstract We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $mathbb{F}_2^nsetminus {vec{0}}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$ . The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $kgeq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-leftlfloor{frac{k}{2^{n-d}}}rightrfloor$ , while for $n gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$ , and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.
{"title":"Subspace coverings with multiplicities","authors":"Anurag Bishnoi, Simona Boyadzhiyska, Shagnik Das, Tamás Mészáros","doi":"10.1017/s0963548323000123","DOIUrl":"https://doi.org/10.1017/s0963548323000123","url":null,"abstract":"Abstract We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $mathbb{F}_2^nsetminus {vec{0}}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$ . The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $kgeq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-leftlfloor{frac{k}{2^{n-d}}}rightrfloor$ , while for $n gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$ , and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135674922","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 : 2023-05-15DOI: 10.1017/s0963548323000135
Fabrizio Frati, Michael Hoffmann, Csaba D. Tóth
Abstract We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class $mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in $mathcal H$ . Our main result is that there exists a geometric graph with $n$ vertices and $O!left(n log nright)$ edges that is universal for $n$ -vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with $n$ vertices and $O!left(n log nright)$ edges that contains every $n$ -vertex forest as a subgraph. The upper bound of $O!left(n log nright)$ edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that every $n$ -vertex convex geometric graph that is universal for $n$ -vertex outerplanar graphs has a near-quadratic number of edges, namely $Omega _h(n^{2-1/h})$ , for every positive integer $h$ ; this almost matches the trivial $O(n^2)$ upper bound given by the $n$ -vertex complete convex geometric graph. Finally, we prove that there exists an $n$ -vertex convex geometric graph with $n$ vertices and $O!left(n log nright)$ edges that is universal for $n$ -vertex caterpillars.
{"title":"Universal geometric graphs","authors":"Fabrizio Frati, Michael Hoffmann, Csaba D. Tóth","doi":"10.1017/s0963548323000135","DOIUrl":"https://doi.org/10.1017/s0963548323000135","url":null,"abstract":"Abstract We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class $mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in $mathcal H$ . Our main result is that there exists a geometric graph with $n$ vertices and $O!left(n log nright)$ edges that is universal for $n$ -vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with $n$ vertices and $O!left(n log nright)$ edges that contains every $n$ -vertex forest as a subgraph. The upper bound of $O!left(n log nright)$ edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that every $n$ -vertex convex geometric graph that is universal for $n$ -vertex outerplanar graphs has a near-quadratic number of edges, namely $Omega _h(n^{2-1/h})$ , for every positive integer $h$ ; this almost matches the trivial $O(n^2)$ upper bound given by the $n$ -vertex complete convex geometric graph. Finally, we prove that there exists an $n$ -vertex convex geometric graph with $n$ vertices and $O!left(n log nright)$ edges that is universal for $n$ -vertex caterpillars.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136215846","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 : 2023-05-02DOI: 10.1017/s0963548323000068
Lior Gishboliner, Benny Sudakov
Abstract A chordal graph is a graph with no induced cycles of length at least $4$ . Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating $f(n,m)$ . In the late 1980s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$ . In this paper we prove this conjecture and answer the question of Erdős and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m leq n^2/3+1$ .
{"title":"Maximal chordal subgraphs","authors":"Lior Gishboliner, Benny Sudakov","doi":"10.1017/s0963548323000068","DOIUrl":"https://doi.org/10.1017/s0963548323000068","url":null,"abstract":"Abstract A chordal graph is a graph with no induced cycles of length at least $4$ . Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating $f(n,m)$ . In the late 1980s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$ . In this paper we prove this conjecture and answer the question of Erdős and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m leq n^2/3+1$ .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135165956","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 : 2023-04-24DOI: 10.1017/s0963548323000044
Zoltán Füredi, András Gyárfás, Zoltán Király
Abstract The notion of cross-intersecting set pair system of size $m$ , $ ({A_i}_{i=1}^m, {B_i}_{i=1}^m )$ with $A_icap B_i=emptyset$ and $A_icap B_jne emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $mlebinom{a+b}{a}$ if $|A_i|le a$ and $|B_i|le b$ for each $i$ . Our central problem is to see how this bound changes with the additional condition $|A_icap B_j|=1$ for $ine j$ . Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is at least $5^{n/2}$ for $n$ even, $a=b=n$ , equal to $bigl (lfloor frac{n}{2}rfloor +1bigr )bigl (lceil frac{n}{2}rceil +1bigr )$ if $a=2$ and $b=nge 4$ , at most $|cup _{i=1}^m A_i|$ , asymptotically $n^2$ if ${A_i}$ is a linear hypergraph ( $|A_icap A_j|le 1$ for $ine j$ ), asymptotically ${1over 2}n^2$ if ${A_i}$ and ${B_i}$ are both linear hypergraphs.
{"title":"Problems and results on 1-cross-intersecting set pair systems","authors":"Zoltán Füredi, András Gyárfás, Zoltán Király","doi":"10.1017/s0963548323000044","DOIUrl":"https://doi.org/10.1017/s0963548323000044","url":null,"abstract":"Abstract The notion of cross-intersecting set pair system of size $m$ , $ ({A_i}_{i=1}^m, {B_i}_{i=1}^m )$ with $A_icap B_i=emptyset$ and $A_icap B_jne emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $mlebinom{a+b}{a}$ if $|A_i|le a$ and $|B_i|le b$ for each $i$ . Our central problem is to see how this bound changes with the additional condition $|A_icap B_j|=1$ for $ine j$ . Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is at least $5^{n/2}$ for $n$ even, $a=b=n$ , equal to $bigl (lfloor frac{n}{2}rfloor +1bigr )bigl (lceil frac{n}{2}rceil +1bigr )$ if $a=2$ and $b=nge 4$ , at most $|cup _{i=1}^m A_i|$ , asymptotically $n^2$ if ${A_i}$ is a linear hypergraph ( $|A_icap A_j|le 1$ for $ine j$ ), asymptotically ${1over 2}n^2$ if ${A_i}$ and ${B_i}$ are both linear hypergraphs.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135223516","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 : 2023-04-24DOI: 10.1017/s0963548323000081
Jie Han, Ping Hu, Guanghui Wang, Donglei Yang
Abstract Given a graph $G$ and an integer $ell ge 2$ , we denote by $alpha _{ell }(G)$ the maximum size of a $K_{ell }$ -free subset of vertices in $V(G)$ . A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$ -vertex graphs $G$ with $alpha _{ell }(G) = o(n)$ , which can be seen as a Ramsey–Turán variant of the celebrated Hajnal–Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$ -factors in $n$ -vertex graphs $G$ with $alpha _ell (G)=n^{1-o(1)}$ for all $rge ell ge 2$ .
给定一个图$G$和一个整数$ell ge 2$，我们用$alpha _{ell }(G)$表示$V(G)$中一个无$K_{ell }$的顶点子集的最大大小。Nenadov和Pehova最近的一个问题是用$alpha _{ell }(G) = o(n)$确定$n$ -顶点图$G$中强迫派系因子的最佳最小可能度条件，这可以看作是著名的hajnal - szemersamedi定理的Ramsey-Turán变体。在本文中，我们找到了$n$ -顶点图$G$中$K_r$ -因子的渐近尖锐最小度阈值，对于所有$rge ell ge 2$都有$alpha _ell (G)=n^{1-o(1)}$。
{"title":"Clique-factors in graphs with sublinear -independence number","authors":"Jie Han, Ping Hu, Guanghui Wang, Donglei Yang","doi":"10.1017/s0963548323000081","DOIUrl":"https://doi.org/10.1017/s0963548323000081","url":null,"abstract":"Abstract Given a graph $G$ and an integer $ell ge 2$ , we denote by $alpha _{ell }(G)$ the maximum size of a $K_{ell }$ -free subset of vertices in $V(G)$ . A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$ -vertex graphs $G$ with $alpha _{ell }(G) = o(n)$ , which can be seen as a Ramsey–Turán variant of the celebrated Hajnal–Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$ -factors in $n$ -vertex graphs $G$ with $alpha _ell (G)=n^{1-o(1)}$ for all $rge ell ge 2$ .","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135277833","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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