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Electronic Journal of Combinatorics最新文献

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Generalized Heawood Numbers 广义希伍德数
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/12104
Wolfgang Kühnel
{"title":"Generalized Heawood Numbers","authors":"Wolfgang Kühnel","doi":"10.37236/12104","DOIUrl":"https://doi.org/10.37236/12104","url":null,"abstract":"This survey explains the origin and the further development of the Heawood inequalities, the Heawood number, and generalizations to higher dimensions with results and further conjectures.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"56 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819669","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 the Maximum $F_5$-Free Subhypergraphs of a Random Hypergraph 关于随机超图的最大$F_5$ free子超图
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/11328
Igor Araujo, József Balogh, Haoran Luo
{"title":"On the Maximum $F_5$-Free Subhypergraphs of a Random Hypergraph","authors":"Igor Araujo, József Balogh, Haoran Luo","doi":"10.37236/11328","DOIUrl":"https://doi.org/10.37236/11328","url":null,"abstract":"Denote by $F_5$ the $3$-uniform hypergraph on vertex set ${1,2,3,4,5}$ with hyperedges ${123,124,345}$. Balogh, Butterfield, Hu, and Lenz proved that if $p > K log n /n$ for some large constant $K$, then every maximum $F_5$-free subhypergraph of $G^3(n,p)$ is tripartite with high probability, and showed that if $p_0 = 0.1sqrt{log n} /n$, then with high probability there exists a maximum $F_5$-free subhypergraph of $G^3(n,p_0)$ that is not tripartite. In this paper, we sharpen the upper bound to be best possible up to a constant factor. We prove that if $p > C sqrt{log n} /n $ for some large constant $C$, then every maximum $F_5$-free subhypergraph of $G^3(n, p)$ is tripartite with high probability.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"51 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869196","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
Three New Refined Arnold Families 三个新的精致阿诺德家族
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/11988
Sen-Peng Eu, Louis Kao
{"title":"Three New Refined Arnold Families","authors":"Sen-Peng Eu, Louis Kao","doi":"10.37236/11988","DOIUrl":"https://doi.org/10.37236/11988","url":null,"abstract":"The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1leq|k|leq n$, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects $(X_{n,k})$ is an Arnold family if $X_{n,k}$ is counted by $v_{n,k}$. A polynomial refinement $V_{n,k}(t)$ of $v_{n,k}$, together with the combinatorial interpretations in several combinatorial structures was introduced by Eu and Fu recently. In this paper, we provide three new Arnold families of combinatorial objects, namely the cycle-up-down permutations, the valley signed permutations and Knuth's flip equivalences on permutations. We shall find corresponding statistics to realize the refined polynomial arrays.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"53 19","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819269","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 Degree and Codegree Threshold for Linear Triangle Covering in 3-Graphs 3-图中线性三角形覆盖的度和余度阈值
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/11717
Yuxuan Tang, Yue Ma, Xinmin Hou
{"title":"The Degree and Codegree Threshold for Linear Triangle Covering in 3-Graphs","authors":"Yuxuan Tang, Yue Ma, Xinmin Hou","doi":"10.37236/11717","DOIUrl":"https://doi.org/10.37236/11717","url":null,"abstract":"Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if every vertex in $G$ is contained in a copy of $F$. For $1le i le k-1$, let $c_i(n,F)$ be the least integer such that every $n$-vertex $k$-uniform hypergraph $G$ with $delta_i(G)> c_i(n,F)$ has an $F$-covering. The covering problem has been systematically studied by Falgas-Ravry and Zhao [Codegree thresholds for covering 3-uniform hypergraphs, [SIAM J. Discrete Math., 2016]. Last year, Falgas-Ravry, Markström, and Zhao [Triangle-degrees in graphs and tetrahedron coverings in 3-graphs, Combinatorics, Probability and Computing, 2021] asymptotically determined $c_1(n, F)$ when $F$ is the generalized triangle. In this note, we give the exact value of $c_2(n, F)$ and asymptotically determine $c_1(n, F)$ when $F$ is the linear triangle $C_6^3$, where $C_6^3$ is the 3-uniform hypergraph with vertex set ${v_1,v_2,v_3,v_4,v_5,v_6}$ and edge set ${v_1v_2v_3,v_3v_4v_5,v_5v_6v_1}$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"55 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819677","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: 2
Irregularity of Graphs Respecting Degree Bounds 关于度界的图的不规则性
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/11948
Dieter Rautenbach, Florian Werner
{"title":"Irregularity of Graphs Respecting Degree Bounds","authors":"Dieter Rautenbach, Florian Werner","doi":"10.37236/11948","DOIUrl":"https://doi.org/10.37236/11948","url":null,"abstract":"Albertson defined the irregularity of a graph $G$ as $$irr(G)=sumlimits_{uvin E(G)}|d_G(u)-d_G(v)|.$$ For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $Delta$, and $d=leftlfloor frac{Delta m}{Delta n-m}rightrfloor$, we show $$irr(G)leq d(d+1)n+frac{1}{Delta}left(Delta^2-(2d+1)Delta-d^2-dright)m.$$","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"52 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135868051","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
Off-Diagonally Symmetric Domino Tilings of the Aztec Diamond 阿兹特克钻石的非对角线对称多米诺骨牌瓷砖
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/11921
Yi-Lin Lee
{"title":"Off-Diagonally Symmetric Domino Tilings of the Aztec Diamond","authors":"Yi-Lin Lee","doi":"10.37236/11921","DOIUrl":"https://doi.org/10.37236/11921","url":null,"abstract":"We introduce a new symmetry class of domino tilings of the Aztec diamond, called the off-diagonal symmetry class, which is motivated by the off-diagonally symmetric alternating sign matrices introduced by Kuperberg in 2002. We use the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths to enumerate our new symmetry class. The number of off-diagonally symmetric domino tilings of the Aztec diamond can be expressed as a Pfaffian of a matrix whose entries satisfy a nice and simple recurrence relation.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"44 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135868865","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 Sequences Without Short Zero-Sum Subsequences 关于没有短零和子序列的序列
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/11963
Xiangneng Zeng, Pingzhi Yuan
{"title":"On Sequences Without Short Zero-Sum Subsequences","authors":"Xiangneng Zeng, Pingzhi Yuan","doi":"10.37236/11963","DOIUrl":"https://doi.org/10.37236/11963","url":null,"abstract":"Let $G$ be a finite abelian group. It is well known that every sequence $S$ over $G$ of length at least $|G|$ contains a zero-sum subsequence of length at most $mathsf{h}(S)$, where $mathsf{h}(S)$ is the maximal multiplicity of elements occurring in $S$. It is interesting to study the corresponding inverse problem, that is to find information on the structure of the sequence $S$ which does not contain zero-sum subsequences of length at most $mathsf{h}(S)$. Under the assumption that $|sum(S)|< min{|G|,2|S|-1}$, Gao, Peng and Wang showed that such a sequence $S$ must be strictly behaving. In the present paper, we explicitly give the structure of such a sequence $S$ under the assumption that $|sum(S)|=2|S|-1<|G|$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"56 20","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819658","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
Weak Degeneracy of Planar Graphs and Locally Planar Graphs 平面图与局部平面图的弱简并
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-11-03 DOI: 10.37236/11749
Ming Han, Tao Wang, Jianglin Wu, Huan Zhou, Xuding Zhu
{"title":"Weak Degeneracy of Planar Graphs and Locally Planar Graphs","authors":"Ming Han, Tao Wang, Jianglin Wu, Huan Zhou, Xuding Zhu","doi":"10.37236/11749","DOIUrl":"https://doi.org/10.37236/11749","url":null,"abstract":"Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d+1)$-list assignment $L$ of $G$, one can construct an $L$ coloring of $G$ by a modified greedy coloring algorithm. It is known that planar graphs of girth 5 are 3-choosable and locally planar graphs are $5$-choosable. This paper strengthens these results and proves that planar graphs of girth 5 are weakly 2-degenerate and locally planar graphs are weakly 4-degenerate.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"56 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819664","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
On Rödl's Theorem for Cographs 关于Rödl图的定理
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-10-20 DOI: 10.37236/12189
Lior Gishboliner, Asaf Shapira
{"title":"On Rödl's Theorem for Cographs","authors":"Lior Gishboliner, Asaf Shapira","doi":"10.37236/12189","DOIUrl":"https://doi.org/10.37236/12189","url":null,"abstract":"A theorem of Rödl states that for every fixed $F$ and $varepsilon&gt;0$ there is $delta=delta_F(varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $delta n$ whose edge density is either at most $varepsilon$ or at least $1-varepsilon$. Rödl's proof relied on the regularity lemma, hence it supplied only a tower-type bound for $delta$. Fox and Sudakov conjectured that $delta$ can be made polynomial in $varepsilon$, and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when $F=P_4$. In fact, they show that the same conclusion holds even if $G$ contains few copies of $P_4$. In this note we give a short proof of a more general statement.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"60 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135568151","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
Ninth Variation of Classical Group Characters of Type A-D and Littlewood Identities A-D型和Littlewood型经典群性状的第九次变异
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-10-20 DOI: 10.37236/11768
Mikhail Goltsblat
{"title":"Ninth Variation of Classical Group Characters of Type A-D and Littlewood Identities","authors":"Mikhail Goltsblat","doi":"10.37236/11768","DOIUrl":"https://doi.org/10.37236/11768","url":null,"abstract":"We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of Foley and King. In this extension, the factorial powers are replaced with an arbitrary sequence of polynomials, as in Sergeev–Veselov's generalised Schur functions and Okada's generalised Schur P- and Q-functions. We also offer a similar generalisation for the rational Schur functions. We derive Littlewood-type identities for our generalisations. These identities allow us to give new (unflagged) Jacobi–Trudi identities for the Foley–King factorial characters and for rational versions of the factorial Schur functions. We also propose an extension of the original Macdonald's ninth variation of Schur functions to the case of symplectic and orthogonal characters, which helps us prove Nägelsbach–Kostka identities.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513629","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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