C. Groenland, Tom Johnston, D'aniel Kor'andi, Alexander Roberts, A. Scott, Jane Tan
Two permutations $s$ and $t$ are $k$-similar if they can be decomposed into subpermutations $s^1, ldots, s^k$ and $t^1, ldots, t^k$ such that $s^i$ is order-isomorphic to $t^i$ for all $i$. Recently, Dudek, Grytczuk and Ruci'nski posed the problem of determining the minimum $k$ for which two permutations chosen independently and uniformly at random are $k$-similar. We show that two such permutations are $O(n^{1/3}log^{11/6}(n))$-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalises to simultaneous decompositions of multiple permutations.
{"title":"Decomposing Random Permutations into Order-Isomorphic Subpermutations","authors":"C. Groenland, Tom Johnston, D'aniel Kor'andi, Alexander Roberts, A. Scott, Jane Tan","doi":"10.1137/22m148029x","DOIUrl":"https://doi.org/10.1137/22m148029x","url":null,"abstract":"Two permutations $s$ and $t$ are $k$-similar if they can be decomposed into subpermutations $s^1, ldots, s^k$ and $t^1, ldots, t^k$ such that $s^i$ is order-isomorphic to $t^i$ for all $i$. Recently, Dudek, Grytczuk and Ruci'nski posed the problem of determining the minimum $k$ for which two permutations chosen independently and uniformly at random are $k$-similar. We show that two such permutations are $O(n^{1/3}log^{11/6}(n))$-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalises to simultaneous decompositions of multiple permutations.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84178113","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}
Pub Date : 2022-02-21DOI: 10.1016/j.disc.2022.112855
A. Woryna
{"title":"On the bijective colouring of Cantor trees based on transducers","authors":"A. Woryna","doi":"10.1016/j.disc.2022.112855","DOIUrl":"https://doi.org/10.1016/j.disc.2022.112855","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76133992","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}
We characterize all digraphs H such that orientations of chordal graphs with no induced copy of H have bounded dichromatic number.
我们刻画了所有有向图H，使得没有诱导拷贝H的弦图的方向具有有界二色数。
{"title":"Heroes in orientations of chordal graphs","authors":"Pierre Aboulker, Guillaume Aubian, R. Steiner","doi":"10.1137/22m1481427","DOIUrl":"https://doi.org/10.1137/22m1481427","url":null,"abstract":"We characterize all digraphs H such that orientations of chordal graphs with no induced copy of H have bounded dichromatic number.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83913697","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}
N. Alon, Anna Gujgiczer, J. Körner, Aleksa Milojević, G. Simonyi
We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.
{"title":"Structured Codes of Graphs","authors":"N. Alon, Anna Gujgiczer, J. Körner, Aleksa Milojević, G. Simonyi","doi":"10.1137/22m1487989","DOIUrl":"https://doi.org/10.1137/22m1487989","url":null,"abstract":"We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85657930","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}
This paper presents new lower bounds for the lattice covering densities of simplices by studying the Degree-Diameter Problem for abelian Cayley digraphs. In particular, it proves that the density of any lattice covering of a tetrahedron is at least $25/18$ and the density of any lattice covering of a four-dimensional simplex is at least $343/264$.
{"title":"Lower Bounds on Lattice Covering Densities of Simplices","authors":"Miao Fu, F. Xue, C. Zong","doi":"10.1137/22m1514155","DOIUrl":"https://doi.org/10.1137/22m1514155","url":null,"abstract":"This paper presents new lower bounds for the lattice covering densities of simplices by studying the Degree-Diameter Problem for abelian Cayley digraphs. In particular, it proves that the density of any lattice covering of a tetrahedron is at least $25/18$ and the density of any lattice covering of a four-dimensional simplex is at least $343/264$.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81904580","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}
For a metric space $(X, d)$ and a scale parameter $r geq 0$, the Vietoris-Rips complex $mathcal{VR}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $sigma subseteq X$ is a simplex if and only if diameter of $sigma$ is at most $r$. For $n geq 1$, let $mathbb{I}_n$ denotes the $n$-dimensional hypercube graph. In this paper, we show that $mathcal{VR}(mathbb{I}_n;r)$ has non trivial reduced homology only in dimensions $4$ and $7$. Therefore, we answer a question posed by Adamaszek and Adams recently. A (finite) simplicial complex $Delta$ is $d$-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most $d$ that is contained in a unique maximal face of $Delta$. The collapsibility number of $Delta$ is the minimum integer $d$ such that $Delta$ is $d$-collapsible. We show that the collapsibility number of $mathcal{VR}(mathbb{I}_n;r)$ is $2^r$ for $r in {2, 3}$.
{"title":"On Vietoris-Rips Complexes (with Scale 3) of Hypercube Graphs","authors":"Samir Shukla","doi":"10.1137/22m1481440","DOIUrl":"https://doi.org/10.1137/22m1481440","url":null,"abstract":"For a metric space $(X, d)$ and a scale parameter $r geq 0$, the Vietoris-Rips complex $mathcal{VR}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $sigma subseteq X$ is a simplex if and only if diameter of $sigma$ is at most $r$. For $n geq 1$, let $mathbb{I}_n$ denotes the $n$-dimensional hypercube graph. In this paper, we show that $mathcal{VR}(mathbb{I}_n;r)$ has non trivial reduced homology only in dimensions $4$ and $7$. Therefore, we answer a question posed by Adamaszek and Adams recently. A (finite) simplicial complex $Delta$ is $d$-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most $d$ that is contained in a unique maximal face of $Delta$. The collapsibility number of $Delta$ is the minimum integer $d$ such that $Delta$ is $d$-collapsible. We show that the collapsibility number of $mathcal{VR}(mathbb{I}_n;r)$ is $2^r$ for $r in {2, 3}$.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75395269","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}
Pub Date : 2022-02-01DOI: 10.1016/j.disc.2021.112679
Sean McGuinness
{"title":"Serial exchanges in matroids","authors":"Sean McGuinness","doi":"10.1016/j.disc.2021.112679","DOIUrl":"https://doi.org/10.1016/j.disc.2021.112679","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84881125","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}
A well-known question of Gallai (1966) asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs, the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex $v$ if and only if it meets the vertex-bond with respect to $v$. Therefore, a very natural approach to Gallai's question is to study whether there is a bond meeting all longest paths. Let $p$ denote the length of a longest path of connected graphs. We show that for any 2-connected graph, there is a bond meeting all paths of length at least $p-1$. We then prove that for any 3-connected graph, there is a bond meeting all paths of length at least $p-2$. For a $k$-connected graph $(kge3)$, we show that there is a bond meeting all paths of length at least $p-t+1$, where $t=Biglfloorsqrt{frac{k-2}{2}}Bigrfloor$ if $p$ is even and $t=Biglceilsqrt{frac{k-2}{2}}Bigrceil$ if $p$ is odd. Our results provide analogs of the corresponding results of P. Wu and S. McGuinness [Bonds intersecting cycles in a graph, Combinatorica 25 (4) (2005), 439-450] also.
{"title":"Bonds Intersecting Long Paths in (k) -Connected Graphs","authors":"Qing-Qing Zhao, B. Wei, Haidong Wu","doi":"10.1137/22m1481105","DOIUrl":"https://doi.org/10.1137/22m1481105","url":null,"abstract":"A well-known question of Gallai (1966) asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs, the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex $v$ if and only if it meets the vertex-bond with respect to $v$. Therefore, a very natural approach to Gallai's question is to study whether there is a bond meeting all longest paths. Let $p$ denote the length of a longest path of connected graphs. We show that for any 2-connected graph, there is a bond meeting all paths of length at least $p-1$. We then prove that for any 3-connected graph, there is a bond meeting all paths of length at least $p-2$. For a $k$-connected graph $(kge3)$, we show that there is a bond meeting all paths of length at least $p-t+1$, where $t=Biglfloorsqrt{frac{k-2}{2}}Bigrfloor$ if $p$ is even and $t=Biglceilsqrt{frac{k-2}{2}}Bigrceil$ if $p$ is odd. Our results provide analogs of the corresponding results of P. Wu and S. McGuinness [Bonds intersecting cycles in a graph, Combinatorica 25 (4) (2005), 439-450] also.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84032347","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}
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets $S$ in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of F"uredi and of Alon, Krivelevich, and Sudakov showing that the maximum number of edges in an $n$-vertex graph not containing a fixed bipartite graph with maximum degree at most $r$ on one side is $O(n^{2-1/r})$. This was recently extended by Grzesik, Janzer and Nagy to the family of so-called $(r,t)$-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov, confirming a special case of a conjecture of ErdH{o}s and Simonovits and of Sidorenko, showing that if $H$ is a bipartite graph that contains a vertex complete to the other part and $G$ is a graph then the probability that the uniform random mapping from $V(H)$ to $V(G)$ is a homomorphismis at least $left[frac{2|E(G)|}{|V(G)|^2}right]^{|E(H)|}$. In this note, we introduce a nested variant of the dependent random choice lemma, which might be of independent interest. We then apply it to obtain a common extension of the theorem of Conlon, Fox, and Sudakov and the theorem of Grzesik, Janzer, and Nagy, regarding Tur'an and Sidorenko properties of so-called tree-degenerate graphs.
著名的依赖随机选择引理指出，在二部图中，一个平均顶点(按其度加权)具有这样的性质，即在其邻域中几乎所有的小子集$S$都具有与具有相同边密度的随机图中几乎一样大的公共邻域。引理的两个著名应用如下。第一个是f redi和Alon、Krivelevich和Sudakov的定理，该定理表明，在不包含最大度为$r$的固定二部图的$n$顶点图中，边的最大个数为$O(n^{2-1/r})$。最近，Grzesik, Janzer和Nagy把这个扩展到所谓的$(r,t)$ -一棵树的膨胀。第二个应用是Conlon, Fox和Sudakov的一个定理，证实了Erd H{o} s和Simonovits以及Sidorenko猜想的一个特例，表明如果$H$是一个包含一个顶点完备于另一部分的二部图，并且$G$是一个图，那么从$V(H)$到$V(G)$的一致随机映射至少$left[frac{2|E(G)|}{|V(G)|^2}right]^{|E(H)|}$是同态的概率。在这篇文章中，我们引入了依赖随机选择引理的一个嵌套变体，它可能会引起独立的兴趣。然后，我们应用它来获得Conlon, Fox, and Sudakov定理和Grzesik, Janzer, and Nagy定理关于Turán和所谓的树退化图的Sidorenko性质的共同扩展。
{"title":"Tree-Degenerate Graphs and Nested Dependent Random Choice","authors":"T. Jiang, Sean Longbrake","doi":"10.1137/22m1483554","DOIUrl":"https://doi.org/10.1137/22m1483554","url":null,"abstract":"The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets $S$ in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of F\"uredi and of Alon, Krivelevich, and Sudakov showing that the maximum number of edges in an $n$-vertex graph not containing a fixed bipartite graph with maximum degree at most $r$ on one side is $O(n^{2-1/r})$. This was recently extended by Grzesik, Janzer and Nagy to the family of so-called $(r,t)$-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov, confirming a special case of a conjecture of ErdH{o}s and Simonovits and of Sidorenko, showing that if $H$ is a bipartite graph that contains a vertex complete to the other part and $G$ is a graph then the probability that the uniform random mapping from $V(H)$ to $V(G)$ is a homomorphismis at least $left[frac{2|E(G)|}{|V(G)|^2}right]^{|E(H)|}$. In this note, we introduce a nested variant of the dependent random choice lemma, which might be of independent interest. We then apply it to obtain a common extension of the theorem of Conlon, Fox, and Sudakov and the theorem of Grzesik, Janzer, and Nagy, regarding Tur'an and Sidorenko properties of so-called tree-degenerate graphs.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87153158","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}
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