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.
本文解释了Heawood不等式、Heawood数的起源和进一步发展，并给出了结果和进一步的猜想。
{"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}
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.
用$F_5$表示顶点集${1,2,3,4,5}$上具有超边${123,124,345}$的$3$ -一致超图。Balogh, Butterfield, Hu, and Lenz证明了如果$p > K log n /n$对于某大常数$K$，那么$G^3(n,p)$的每一个极大$F_5$自由子超图都有大概率是三方的，并且证明了如果$p_0 = 0.1sqrt{log n} /n$，那么有大概率存在$G^3(n,p_0)$的极大$F_5$自由子超图不是三方的。在本文中，我们锐化上界，使其尽可能达到一个常数因子。证明了如果$p > C sqrt{log n} /n $对于某大常数$C$，则$G^3(n, p)$的每一个极大$F_5$自由子超图都是高概率的三部。
{"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}
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.
{"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}
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 ${v_1v_2v_3,v_3v_4v_5,v_5v_6v_1}$.
给定两个$k$ -一致超图$F$和$G$，如果$G$中的每个顶点都包含在$F$的副本中，那么我们说$G$有一个$F$ -覆盖。对于$1le i le k-1$，设$c_i(n,F)$为最小整数，使得每个$n$ -顶点$k$ -均匀超图$G$与$delta_i(G)> c_i(n,F)$都有一个$F$ -覆盖。Falgas-Ravry和Zhao[覆盖3-均匀超图的共度阈值，[j]离散数学。[j]。去年，Falgas-Ravry, Markström和Zhao[图中的三角形度和3-图中的四面体覆盖，组合学，概率与计算，2021]渐近确定$c_1(n, F)$时$F$是广义三角形。在这篇文章中，我们给出了$c_2(n, F)$的确切值并渐近地确定$c_1(n, F)$，当$F$是线性三角形$C_6^3$，其中$C_6^3$是顶点集${v_1,v_2,v_3,v_4,v_5,v_6}$和边集${v_1v_2v_3,v_3v_4v_5,v_5v_6v_1}$的3-均匀超图。
{"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}
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 $$irr(G)leq d(d+1)n+frac{1}{Delta}left(Delta^2-(2d+1)Delta-d^2-dright)m.$$
{"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}
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.
{"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}
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 $|sum(S)|=2|S|-1<|G|$.
{"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}
Ming Han, Tao Wang, Jianglin Wu, Huan Zhou, Xuding Zhu
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.
{"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}
A theorem of Rödl states that for every fixed $F$ and $varepsilon>0$ 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.
{"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 $varepsilon>0$ 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}
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.
{"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}
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