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On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers 论可嵌入超立方体一层的图及其极值数
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-07-29 DOI: 10.1007/s00026-024-00705-2
Maria Axenovich, Ryan R. Martin, Christian Winter
{"title":"On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers","authors":"Maria Axenovich,&nbsp;Ryan R. Martin,&nbsp;Christian Winter","doi":"10.1007/s00026-024-00705-2","DOIUrl":"10.1007/s00026-024-00705-2","url":null,"abstract":"<div><p>A graph is cubical if it is a subgraph of a hypercube. For a cubical graph <i>H</i> and a hypercube <span>(Q_n)</span>, <span>(textrm{ex}(Q_n, H))</span> is the largest number of edges in an <i>H</i>-free subgraph of <span>(Q_n)</span>. If <span>(textrm{ex}(Q_n, H))</span> is at least a positive proportion of the number of edges in <span>(Q_n)</span>, then <i>H</i> is said to have positive Turán density in the hypercube; otherwise it has zero Turán density. Determining <span>(textrm{ex}(Q_n, H))</span> and even identifying whether <i>H</i> has positive or zero Turán density remains a widely open question for general <i>H</i>. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge layer of some hypercube. Graphs <i>H</i> that are not layered have positive Turán density because one can form an <i>H</i>-free subgraph of <span>(Q_n)</span> consisting of edges of every other layer. For example, a 4-cycle is not layered and has positive Turán density. However, in general, it is not obvious what properties layered graphs have. We give a characterization of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least 12 and of length 8. However, we prove that there are cubical graphs of girth 8 that are not layered and thus having positive Turán density. The cycle of length 10 remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that <span>(textrm{ex}(Q_n, C_{10})= Omega (n2^n/ log ^a n))</span>, for a constant <i>a</i>, showing that the extremal number for a 10-cycle behaves differently from any other cycle of zero Turán density.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1257 - 1283"},"PeriodicalIF":0.6,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00705-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Likely Maximum Size of Twin Subtrees in a Large Random Tree 大型随机树中双子树可能的最大尺寸
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-07-27 DOI: 10.1007/s00026-024-00711-4
Miklós Bóna, Ovidiu Costin, Boris Pittel
{"title":"The Likely Maximum Size of Twin Subtrees in a Large Random Tree","authors":"Miklós Bóna,&nbsp;Ovidiu Costin,&nbsp;Boris Pittel","doi":"10.1007/s00026-024-00711-4","DOIUrl":"10.1007/s00026-024-00711-4","url":null,"abstract":"<div><p>We call a pair of vertex-disjoint, induced subtrees of a rooted tree twins if they have the same counts of vertices by out-degrees. The likely maximum size of twins in a uniformly random, rooted Cayley tree of size <span>(nrightarrow infty )</span> is studied. It is shown that the expected number of twins of size <span>((2+delta )sqrt{log ncdot log log n})</span> approaches zero, while the expected number of twins of size <span>((2-delta )sqrt{log ncdot log log n})</span> approaches infinity.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"167 - 181"},"PeriodicalIF":0.6,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771052","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
Symmetry Parameters of Two-Generator Circulant Graphs 双发电机圆周图的对称参数
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-07-26 DOI: 10.1007/s00026-024-00709-y
Sally Cockburn, Sarah Loeb
{"title":"Symmetry Parameters of Two-Generator Circulant Graphs","authors":"Sally Cockburn,&nbsp;Sarah Loeb","doi":"10.1007/s00026-024-00709-y","DOIUrl":"10.1007/s00026-024-00709-y","url":null,"abstract":"<div><p>The derived graph of a voltage graph consisting of a single vertex and two loops of different voltages is a circulant graph with two generators. We characterize the automorphism groups of connected, two-generator circulant graphs, and give their determining and distinguishing number, and when relevant, their cost of 2-distinguishing. We do the same for the subdivisions of connected, two-generator circulant graphs obtained by replacing one loop in the voltage graph with a directed cycle.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1093 - 1117"},"PeriodicalIF":0.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771034","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
The Prym Variety of a Dilated Double Cover of Metric Graphs 公元图的扩张双覆盖的普里姆变体
IF 0.7 4区 数学
Annals of Combinatorics Pub Date : 2024-07-08 DOI: 10.1007/s00026-024-00707-0
Arkabrata Ghosh, Dmitry Zakharov
{"title":"The Prym Variety of a Dilated Double Cover of Metric Graphs","authors":"Arkabrata Ghosh,&nbsp;Dmitry Zakharov","doi":"10.1007/s00026-024-00707-0","DOIUrl":"10.1007/s00026-024-00707-0","url":null,"abstract":"<div><p>We calculate the volume of the tropical Prym variety of a harmonic double cover of metric graphs having non-trivial dilation. We show that the tropical Prym variety behaves discontinuously under deformations of the double cover that change the number of connected components of the dilation subgraph.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 2","pages":"439 - 460"},"PeriodicalIF":0.7,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570620","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 Bump Statistic on Permutations Resulting from the Robinson–Schensted Correspondence 罗宾逊-申斯泰德对应关系产生的排列的凹凸统计量
IF 0.7 4区 数学
Annals of Combinatorics Pub Date : 2024-07-08 DOI: 10.1007/s00026-024-00708-z
Mark Dukes, Andrew Mullins
{"title":"A Bump Statistic on Permutations Resulting from the Robinson–Schensted Correspondence","authors":"Mark Dukes,&nbsp;Andrew Mullins","doi":"10.1007/s00026-024-00708-z","DOIUrl":"10.1007/s00026-024-00708-z","url":null,"abstract":"<div><p>In this paper, we investigate a permutation statistic that was independently introduced by Romik (Funct Anal Appl 39(2):52–155, 2005). This statistic counts the number of bumps that occur during the execution of the Robinson–Schensted procedure when applied to a given permutation. We provide several interpretations of this bump statistic that include the tableaux shape and also as an extremal problem concerning permutations and increasing subsequences. Several aspects of this bump statistic are investigated from both structural and enumerative viewpoints.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 2","pages":"375 - 394"},"PeriodicalIF":0.7,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00708-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Positivity of Infinite Products Connected to Partitions with Even Parts Below Odd Parts and Copartitions 论与偶数部分低于奇数部分的分区和共分区相连的无限积的实在性
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-07-01 DOI: 10.1007/s00026-024-00704-3
Hannah E. Burson, Dennis Eichhorn
{"title":"On the Positivity of Infinite Products Connected to Partitions with Even Parts Below Odd Parts and Copartitions","authors":"Hannah E. Burson,&nbsp;Dennis Eichhorn","doi":"10.1007/s00026-024-00704-3","DOIUrl":"10.1007/s00026-024-00704-3","url":null,"abstract":"<div><p>In this paper, we give a combinatorial proof of a positivity result of Chern related to Andrews’s <span>(mathcal{E}mathcal{O}^*)</span>-type partitions. This combinatorial proof comes after reframing Chern’s result in terms of copartitions.Using this new perspective, we also reprove an overpartition result of Chern by showing that it comes essentially “for free” from our combinatorial proof and some basic properties of copartitions. Finally, the application of copartitions leads us to more general positivity conjectures for families of both infinite and finite products, with a proof in one special case.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"197 - 210"},"PeriodicalIF":0.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509563","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
Insertion Algorithms for Gelfand (S_n)-Graphs 格尔凡$S_n$$图的插入算法
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-06-28 DOI: 10.1007/s00026-024-00701-6
Eric Marberg, Yifeng Zhang
{"title":"Insertion Algorithms for Gelfand (S_n)-Graphs","authors":"Eric Marberg,&nbsp;Yifeng Zhang","doi":"10.1007/s00026-024-00701-6","DOIUrl":"10.1007/s00026-024-00701-6","url":null,"abstract":"<div><p>The two tableaux assigned by the Robinson–Schensted correspondence are equal if and only if the input permutation is an involution, so the RS algorithm restricts to a bijection between involutions in the symmetric group and standard tableaux. Beissinger found a concise way of formulating this restricted map, which involves adding an extra cell at the end of a row after a Schensted insertion process. We show that by changing this algorithm slightly to add cells at the end of columns rather than rows, one obtains a different bijection from involutions to standard tableaux. Both maps have an interesting connection to representation theory. Specifically, our insertion algorithms classify the molecules (and conjecturally the cells) in the pair of <i>W</i>-graphs associated with the unique equivalence class of perfect models for a generic symmetric group.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1199 - 1242"},"PeriodicalIF":0.6,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00701-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Legendre Theorems for a Class of Partitions with Initial Repetitions 具有初始重复的一类分区的勒让德定理
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-06-25 DOI: 10.1007/s00026-024-00706-1
Darlison Nyirenda, Beaullah Mugwangwavari
{"title":"Legendre Theorems for a Class of Partitions with Initial Repetitions","authors":"Darlison Nyirenda, Beaullah Mugwangwavari","doi":"10.1007/s00026-024-00706-1","DOIUrl":"https://doi.org/10.1007/s00026-024-00706-1","url":null,"abstract":"<p>Partitions with initial repetitions were introduced by George Andrews. We consider a subclass of these partitions and find Legendre theorems associated with their respective partition functions. The results in turn provide partition-theoretic interpretations of some Rogers–Ramanujan identities due to Lucy J. Slater.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"73 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509565","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
Fan Valuations and Spherical Intrinsic Volumes 扇形估值和球形内在体积
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-06-24 DOI: 10.1007/s00026-024-00699-x
Spencer Backman, Sebastian Manecke, Raman Sanyal
{"title":"Fan Valuations and Spherical Intrinsic Volumes","authors":"Spencer Backman,&nbsp;Sebastian Manecke,&nbsp;Raman Sanyal","doi":"10.1007/s00026-024-00699-x","DOIUrl":"10.1007/s00026-024-00699-x","url":null,"abstract":"<div><p>We generalize valuations on polyhedral cones to valuations on (plane) fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion– restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion–restriction proof of a result of Klivans–Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these <i>intrinsic indicators</i> yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1285 - 1302"},"PeriodicalIF":0.6,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509566","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
Infinite Families of Vertex-Transitive Graphs with Prescribed Hamilton Compression 具有规定汉密尔顿压缩的顶点-传递图无穷族
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-06-20 DOI: 10.1007/s00026-024-00703-4
Klavdija Kutnar, Dragan Marušič, Andriaherimanana Sarobidy Razafimahatratra
{"title":"Infinite Families of Vertex-Transitive Graphs with Prescribed Hamilton Compression","authors":"Klavdija Kutnar,&nbsp;Dragan Marušič,&nbsp;Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1007/s00026-024-00703-4","DOIUrl":"10.1007/s00026-024-00703-4","url":null,"abstract":"<div><p>Given a graph <i>X</i> with a Hamilton cycle <i>C</i>, the <i>compression factor </i><span>(kappa (X,C))</span> <i>of </i><i>C</i> is the order of the largest cyclic subgroup of <span>({textrm{Aut}},(C)cap {textrm{Aut}},(X))</span>, and the <i>Hamilton compression </i><span>(kappa (X))</span> <i>of </i><i>X</i> is the maximum of <span>(kappa (X,C))</span> where <i>C</i> runs over all Hamilton cycles in <i>X</i>. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor et al. (Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y, 2023) whether for every positive integer <i>k</i>, there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to <i>k</i>. Since an infinite family of Cayley graphs with Hamilton compression equal to 1 was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products <span>(mathbb {Z}_prtimes mathbb {Z}_k)</span> where <i>p</i> is a prime and <span>(k ge 2)</span> a divisor of <span>(p-1)</span>. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to 1 are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given.\u0000</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1243 - 1255"},"PeriodicalIF":0.6,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509568","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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