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Electronic Journal of Combinatorics最新文献

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Rank-Metric Lattices Rank-Metric晶格
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-01-13 DOI: 10.37236/11373
Giuseppe Cotardo, Alberto Ravagnani
{"title":"Rank-Metric Lattices","authors":"Giuseppe Cotardo, Alberto Ravagnani","doi":"10.37236/11373","DOIUrl":"https://doi.org/10.37236/11373","url":null,"abstract":"We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polynomiality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135897990","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
Extremal Independent Set Reconfiguration 极值独立集重构
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-01-05 DOI: 10.48550/arXiv.2301.02020
N. Bousquet, Bastien Durain, Théo Pierron, St'ephan Thomass'e
{"title":"Extremal Independent Set Reconfiguration","authors":"N. Bousquet, Bastien Durain, Théo Pierron, St'ephan Thomass'e","doi":"10.48550/arXiv.2301.02020","DOIUrl":"https://doi.org/10.48550/arXiv.2301.02020","url":null,"abstract":"The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on independent sets are widely studied: for example, it is well known that an $n$-vertex graph has at most $3^{n/3}$ maximum independent sets (and this is tight). This paper investigates the asymptotic behavior of maximum possible length of a shortest reconfiguration sequence for independent sets of size $k$ among all $n$-vertex graphs. We give a tight bound for $k=2$. We also provide a subquadratic upper bound (using the hypergraph removal lemma) as well as an almost tight construction for $k=3$. We generalize our results for larger values of $k$ by proving an $n^{2lfloor k/3 rfloor}$ lower bound.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"PP 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84163349","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
Rowmotion on 321-Avoiding Permutations 关于321-避免排列的动议
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-21 DOI: 10.37236/11792
Ben Adenbaum, S. Elizalde
{"title":"Rowmotion on 321-Avoiding Permutations","authors":"Ben Adenbaum, S. Elizalde","doi":"10.37236/11792","DOIUrl":"https://doi.org/10.37236/11792","url":null,"abstract":"We give a natural definition of rowmotion for $321$-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne--Kreweras involution, the analogous notion for antichains of the positive root poset of type $A$. We prove that some permutation statistics, such as the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. \u0000Our setting also provides a more natural description of the celebrated Armstrong--Stump--Thomas equivariant bijection between antichains and non-crossing matchings in types $A$ and $B$, by showing that it is equivalent to the Robinson--Schensted--Knuth correspondence on $321$-avoiding permutations permutations.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"389 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80796629","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: 3
A Linear Hypergraph Extension of Turán's Theorem Turán定理的线性超图推广
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-16 DOI: 10.37236/10525
Guorong Gao, A. Chang
{"title":"A Linear Hypergraph Extension of Turán's Theorem","authors":"Guorong Gao, A. Chang","doi":"10.37236/10525","DOIUrl":"https://doi.org/10.37236/10525","url":null,"abstract":"An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of $r$-uniform hypergraphs $mathcal{F}$, the linear Turán number ex$_r^{lin}(n,mathcal{F})$ is the maximum number of edges of a linear $r$-uniform hypergraph on $n$ vertices that does not contain any member of $mathcal{F}$ as a subgraph. \u0000Let $K_l$ be a complete graph with $l$ vertices and $rgeq 2$. The $r$-expansion of $K_l$ is the $r$-graph $K_l^+$ obtained from $K_l$ by enlarging each edge of $K_l$ with $r-2$ new vertices disjoint from $V(K_l)$ such that distinct edges of $K_l$ are enlarged by distinct vertices. When $lgeq r geq 3$ and $n$ is sufficiently large, we prove the following extension of Turán's Theorem $$ex_{r}^{lin}left(n, K_{l+1}^{+}right)leq |TD_r(n,l)|,$$ with equality achieved only by the Turán design $TD_r(n,l)$, where the Turán design $TD_r(n,l)$ is an almost balanced $l$-partite $r$-graph such that each pair of vertices from distinct parts are contained in one edge exactly. Moreover, some results on linear Turán number of general configurations are also presented.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87609445","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 Combinatorial Characterization of Extremal Generalized Hexagons 极值广义六边形的组合表征
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-16 DOI: 10.37236/9245
B. Bruyn
{"title":"A Combinatorial Characterization of Extremal Generalized Hexagons","authors":"B. Bruyn","doi":"10.37236/9245","DOIUrl":"https://doi.org/10.37236/9245","url":null,"abstract":"A finite generalized $2d$-gon of order $(s,t)$ with $d in { 2,3,4 }$ and $s not= 1$ is called extremal if $t$ attains its maximal possible value $s^{e_d}$, where $e_2=e_4=2$ and $e_3=3$. The problem of finding combinatorial conditions that are both necessary and sufficient for a finite generalized $2d$-gon of order $(s,t)$ to be extremal has so far only be solved for the generalized quadrangles. In this paper, we obtain a solution for the generalized hexagons. We also obtain a related combinatorial characterization for extremal regular near hexagons.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"5 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75390365","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
Density of Balanced 3-Partite Graphs without 3-Cycles or 4-Cycles 无3环或4环的平衡3部图的密度
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-16 DOI: 10.37236/10958
Zequn Lv, Mei Lu, Chunqiu Fang
{"title":"Density of Balanced 3-Partite Graphs without 3-Cycles or 4-Cycles","authors":"Zequn Lv, Mei Lu, Chunqiu Fang","doi":"10.37236/10958","DOIUrl":"https://doi.org/10.37236/10958","url":null,"abstract":"Let $C_k$ be a cycle of order $k$, where $kge 3$. Let ex$(n, n, n, {C_{3}, C_{4}})$ be the maximum number of edges in a balanced $3$-partite graph whose vertex set consists of three parts, each has $n$ vertices that has no subgraph isomorphic to $C_3$ or $C_4$. We construct dense balanced 3-partite graphs without 3-cycles or 4-cycles and show that ex$(n, n, n, {C_{3}, C_{4}})ge (frac{6sqrt{2}-8}{(sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82731889","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
Weighted Modulo Orientations of Graphs and Signed Graphs 图与符号图的加权模取向
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-16 DOI: 10.37236/10740
Jianbing Liu, Miaomiao Han, H. Lai
{"title":"Weighted Modulo Orientations of Graphs and Signed Graphs","authors":"Jianbing Liu, Miaomiao Han, H. Lai","doi":"10.37236/10740","DOIUrl":"https://doi.org/10.37236/10740","url":null,"abstract":"Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) to {mathbb Z}_psetminus{0}$ and a ${mathbb Z}_p$-boundary $b$ of $G$, an orientation $tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${mathbb Z}_p$ at each vertex $vin V(G)$ under orientation $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that $(6p^2 - 14p + 8)$-edge-connected graphs have all possible $(f,b;p)$-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the $(f,b;p)$-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87094159","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
Splitting Matchings and the Ryser-Brualdi-Stein Conjecture for Multisets 多集的分裂匹配和Ryser-Brualdi-Stein猜想
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-06 DOI: 10.37236/11714
Michael Anastos, David Fabian, Alp Müyesser, T. Szab'o
{"title":"Splitting Matchings and the Ryser-Brualdi-Stein Conjecture for Multisets","authors":"Michael Anastos, David Fabian, Alp Müyesser, T. Szab'o","doi":"10.37236/11714","DOIUrl":"https://doi.org/10.37236/11714","url":null,"abstract":"We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3in mathbb{N}$ with $a_1+a_2+a_3leq n-2$, we show there exists a matching $M$ of $G$ with $|Mcap M_i|=a_i$ for each $iin {1,2,3}$. The bound $n-2$ in the theorem is best possible in general.We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"458 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75102745","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
On Infinite, Cubic, Vertex-Transitive Graphs with Applications to Totally Disconnected, Locally Compact Groups 无限三次顶点传递图及其在完全不连通局部紧群上的应用
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-02 DOI: 10.37236/10709
Arnbjörg Soffía Árnadóttir, Waltraud Lederle, R. G. Möller
{"title":"On Infinite, Cubic, Vertex-Transitive Graphs with Applications to Totally Disconnected, Locally Compact Groups","authors":"Arnbjörg Soffía Árnadóttir, Waltraud Lederle, R. G. Möller","doi":"10.37236/10709","DOIUrl":"https://doi.org/10.37236/10709","url":null,"abstract":"We study groups acting vertex-transitively and non-discretely on connected, cubic graphs (regular graphs of degree 3). Using ideas from Tutte's fundamental papers in 1947 and 1959, it is shown that if the action is edge-transitive, then the graph has to be a tree. When the action is not edge-transitive Tutte's ideas are still useful and can, amongst other things, be used to fully classify the possible two-ended graphs. Results about cubic graphs are then applied to Willis' scale function from the theory of totally disconnected, locally compact groups. Some of the results in this paper have most likely been known to experts but most of them are not stated explicitly with proofs in the literature.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"140 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82103296","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
Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics 用曲柄统计量研究高斯多项式连续系数的同余性
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2022-12-02 DOI: 10.37236/10493
Dennis Eichhorn, Lydia Engle, Brandt Kronholm
{"title":"Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics","authors":"Dennis Eichhorn, Lydia Engle, Brandt Kronholm","doi":"10.37236/10493","DOIUrl":"https://doi.org/10.37236/10493","url":null,"abstract":"\u0000 \u0000 \u0000In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime $ell$ for the function $pbig(n,m,Nbig)$, which enumerates the partitions of $n$ into at most $m$ parts with no part larger than $N$. We also treat the function $pbig(n,m,(a,b]big)$, which bounds the largest part above and below, and obtain similar infinite families of congruences. \u0000For $m leq 4$ and $ell = 3$, simple combinatorial statistics called \"cranks\" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons. \u0000  \u0000For $m leq 4$ and $ell = 3$, simple combinatorial statistics called ``cranks\"  witness these congruences.  We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$.  Our combinatorial proof relies upon explicit dissections of convex lattice polygons.  \u0000 \u0000 \u0000","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"8 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75679116","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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