Pub Date : 2018-04-01DOI: 10.4310/JOC.2019.V10.N3.A1
B. Rhoades, A. Wilson
A set partition of $[n] := {1, 2, dots, n }$ is called {em $r$-Stirling} if the numbers $1, 2, dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r leq k leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of the {em coinversion code} of an ordered set partition to reprove and generalize some results of Haglund et. al. in a more direct way. Furthermore, we introduce a variety $X_{n,k}^{(r)}$ of line arrangements whose cohomology is presented as the integral form of $R_{n,k}^{(r)}$, generalizing results of Pawlowski and Rhoades.
如果数字$1, 2, dots, r$属于不同的块，则将{em}$[n] := {1, 2, dots, n }${em的集合分区称为}{em$r$} -Stirling。Haglund, Rhoades和Shimozono根据两个正整数$k leq n$构造了梯度环$R_{n,k}$，这两个正整数的代数性质由$[n]$与$k$块的有序集划分的组合控制。对于依赖于三个整数$r leq k leq n$的有序$r$ -Stirling分区，我们引入了这个商的一个变体$R_{n,k}^{(r)}$。我们描述了$R_{n,k}^{(r)}$的标准单项式基，并利用有序集划分的{em共反演码}的组合概念，更直接地修正和推广了Haglund等人的一些结果。进一步，我们引入了各种$X_{n,k}^{(r)}$的线排列，它们的上同调被表示为$R_{n,k}^{(r)}$的积分形式，推广了Pawlowski和Rhoades的结果。
{"title":"Line configurations and $r$-Stirling partitions","authors":"B. Rhoades, A. Wilson","doi":"10.4310/JOC.2019.V10.N3.A1","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N3.A1","url":null,"abstract":"A set partition of $[n] := {1, 2, dots, n }$ is called {em $r$-Stirling} if the numbers $1, 2, dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r leq k leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of the {em coinversion code} of an ordered set partition to reprove and generalize some results of Haglund et. al. in a more direct way. Furthermore, we introduce a variety $X_{n,k}^{(r)}$ of line arrangements whose cohomology is presented as the integral form of $R_{n,k}^{(r)}$, generalizing results of Pawlowski and Rhoades.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"4 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74699477","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 : 2018-03-02DOI: 10.4310/JOC.2019.V10.N3.A3
Gi-Sang Cheon, Jinha Kim, Minki Kim, S. Kitaev, A. Pyatkin
Distinct letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word of the form $xyxycdots$ (of even or odd length) or a word of the form $yxyxcdots$ (of even or odd length). A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. Thus, edges of $G$ are defined by avoiding the consecutive pattern 11 in a word representing $G$, that is, by avoiding $xx$ and $yy$. In 2017, Jeff Remmel has introduced the notion of a $k$-11-representable graph for a non-negative integer $k$, which generalizes the notion of a word-representable graph. Under this representation, edges of $G$ are defined by containing at most $k$ occurrences of the consecutive pattern 11 in a word representing $G$. Thus, word-representable graphs are precisely $0$-11-representable graphs. In this paper, we study properties of $k$-11-representable graphs for $kgeq 1$, in particular, showing that the class of word-representable graphs, studied intensively in the literature, is contained strictly in the class of 1-11-representable graphs. Another particular result that we prove is the fact that the class of interval graphs is precisely the class of 1-11-representable graphs that can be represented by uniform words containing two copies of each letter. This result can be compared with the known fact that the class of circle graphs is precisely the class of 0-11-representable graphs that can be represented by uniform words containing two copies of each letter. Also, one of our key results in this paper is the fact that any graph is $k$-11-representable for some $kgeq 0$.
{"title":"On $ktextrm{-}11$-representable graphs","authors":"Gi-Sang Cheon, Jinha Kim, Minki Kim, S. Kitaev, A. Pyatkin","doi":"10.4310/JOC.2019.V10.N3.A3","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N3.A3","url":null,"abstract":"Distinct letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word of the form $xyxycdots$ (of even or odd length) or a word of the form $yxyxcdots$ (of even or odd length). A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. Thus, edges of $G$ are defined by avoiding the consecutive pattern 11 in a word representing $G$, that is, by avoiding $xx$ and $yy$. \u0000In 2017, Jeff Remmel has introduced the notion of a $k$-11-representable graph for a non-negative integer $k$, which generalizes the notion of a word-representable graph. Under this representation, edges of $G$ are defined by containing at most $k$ occurrences of the consecutive pattern 11 in a word representing $G$. Thus, word-representable graphs are precisely $0$-11-representable graphs. In this paper, we study properties of $k$-11-representable graphs for $kgeq 1$, in particular, showing that the class of word-representable graphs, studied intensively in the literature, is contained strictly in the class of 1-11-representable graphs. Another particular result that we prove is the fact that the class of interval graphs is precisely the class of 1-11-representable graphs that can be represented by uniform words containing two copies of each letter. This result can be compared with the known fact that the class of circle graphs is precisely the class of 0-11-representable graphs that can be represented by uniform words containing two copies of each letter. Also, one of our key results in this paper is the fact that any graph is $k$-11-representable for some $kgeq 0$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"1997 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88189461","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 : 2018-02-09DOI: 10.4310/joc.2020.v11.n2.a1
Aaron Robertson
Let $r$ and $k$ be positive integers with $r mid k$. Denote by $w_{mathrm{mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $chi:[1,w_{mathrm{mathfrak{z}}}(k;r)] rightarrow {0,1,dots,r-1}$ admits a $k$-term arithmetic progression $a,a+d,dots,a+(k-1)d$ with $sum_{j=0}^{k-1} chi(a+jd) equiv 0 ,(mathrm{mod },r)$. We investigate these numbers as well as a "mixed" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{mathrm{mathfrak{z}}}(k;r)$.
{"title":"Zero-sum analogues of van der Waerden’s theorem on arithmetic progressions","authors":"Aaron Robertson","doi":"10.4310/joc.2020.v11.n2.a1","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n2.a1","url":null,"abstract":"Let $r$ and $k$ be positive integers with $r mid k$. Denote by $w_{mathrm{mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $chi:[1,w_{mathrm{mathfrak{z}}}(k;r)] rightarrow {0,1,dots,r-1}$ admits a $k$-term arithmetic progression $a,a+d,dots,a+(k-1)d$ with $sum_{j=0}^{k-1} chi(a+jd) equiv 0 ,(mathrm{mod },r)$. We investigate these numbers as well as a \"mixed\" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{mathrm{mathfrak{z}}}(k;r)$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"2 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78664494","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 : 2018-01-23DOI: 10.4310/JOC.2019.V10.N4.A2
A. Garsia, J. Liese, J. Remmel, Meesue Yoo
In cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $Delta_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in cite{GHRY} that the Delta Conjecture is true when either $q=0$ or $t=0$. In this paper we complete a work initiated by Remmel whose initial aim was to explore the symmetric function $Delta_{s_nu} e_n$ by the same methods developed in cite{GHRY}. Our first need here is a method for constructing a symmetric function that may be viewed as a "combinatorial side" for the symmetric function $Delta_{s_nu} e_n$ for $t=0$. Based on what was discovered in cite{GHRY} we conjectured such a construction mechanism. We prove here that in the case that $nu=(m-k,1^k)$ with $1le m< n$ the equality of the two sides can be established by the same methods used in cite{GHRY}. While this work was in progress, we learned that Rhodes and Shimozono had previously constructed also such a "combinatorial side". Very recently, Jim Haglund was able to prove that their conjecture follows from the results in cite{GHRY}. We show here that an appropriate modification of the Haglund arguments proves that the polynomial $Delta_{s_nu}e_n$ as well as the Rhoades-Shimozono "combinatorial side" have a plethystic evaluation with hook Schur function expansion.
{"title":"Exploring a Delta Schur Conjecture","authors":"A. Garsia, J. Liese, J. Remmel, Meesue Yoo","doi":"10.4310/JOC.2019.V10.N4.A2","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N4.A2","url":null,"abstract":"In cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $Delta_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in cite{GHRY} that the Delta Conjecture is true when either $q=0$ or $t=0$. In this paper we complete a work initiated by Remmel whose initial aim was to explore the symmetric function $Delta_{s_nu} e_n$ by the same methods developed in cite{GHRY}. Our first need here is a method for constructing a symmetric function that may be viewed as a \"combinatorial side\" for the symmetric function $Delta_{s_nu} e_n$ for $t=0$. Based on what was discovered in cite{GHRY} we conjectured such a construction mechanism. We prove here that in the case that $nu=(m-k,1^k)$ with $1le m< n$ the equality of the two sides can be established by the same methods used in cite{GHRY}. While this work was in progress, we learned that Rhodes and Shimozono had previously constructed also such a \"combinatorial side\". Very recently, Jim Haglund was able to prove that their conjecture follows from the results in cite{GHRY}. We show here that an appropriate modification of the Haglund arguments proves that the polynomial $Delta_{s_nu}e_n$ as well as the Rhoades-Shimozono \"combinatorial side\" have a plethystic evaluation with hook Schur function expansion.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84760217","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 : 2018-01-21DOI: 10.4310/JOC.2021.v12.n2.a6
G. Gutowski, Tomasz Krawczyk, Krzysztof Maziarz, D. West, Michal Zajkac, Xuding Zhu
The emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $frac73n$ when $G$ is outerplanar.
{"title":"The slow-coloring game on sparse graphs: $k$-degenerate, planar, and outerplanar","authors":"G. Gutowski, Tomasz Krawczyk, Krzysztof Maziarz, D. West, Michal Zajkac, Xuding Zhu","doi":"10.4310/JOC.2021.v12.n2.a6","DOIUrl":"https://doi.org/10.4310/JOC.2021.v12.n2.a6","url":null,"abstract":"The emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $frac73n$ when $G$ is outerplanar.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"218 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76969642","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 : 2017-12-23DOI: 10.4310/JOC.2019.v10.n3.a4
C. Benedetti, N. Bergeron, John M. Machacek
In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a characterization of hypergraphs giving rise to simple hypergraphic polytopes in terms of acyclic orientations of the hypergraph. In particular, we recover this fact for the nestohedra and the hyper-permutahedra, and prove it for generalized Pitman-Stanley polytopes as defined here.
{"title":"Hypergraphic polytopes: combinatorial properties and antipode","authors":"C. Benedetti, N. Bergeron, John M. Machacek","doi":"10.4310/JOC.2019.v10.n3.a4","DOIUrl":"https://doi.org/10.4310/JOC.2019.v10.n3.a4","url":null,"abstract":"In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs and faces of hypergraphic polytopes. This allows us to obtain a geometric interpretation of the coefficients of the antipode map in a Hopf algebra of hypergraphs. This interpretation differs from similar ones for a different Hopf structure on hypergraphs provided recently by Aguiar and Ardila. Furthermore, making use of the tools and definitions developed here regarding orientations of hypergraphs we provide a characterization of hypergraphs giving rise to simple hypergraphic polytopes in terms of acyclic orientations of the hypergraph. In particular, we recover this fact for the nestohedra and the hyper-permutahedra, and prove it for generalized Pitman-Stanley polytopes as defined here.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"70 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77398479","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 : 2017-12-04DOI: 10.4310/JOC.2020.V11.N4.A2
Oliver Cooley, Abraham Guti'errez
The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle. It can also be seen as a measure of whether two graphs on a common vertex set are "jointly connected". In this paper we consider the natural generalisation of this process to an arbitrary number of graphs on the same vertex set. We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities. This generalises a result of Bollobas, Riordan, Slivken and Smith.
{"title":"Multi-coloured jigsaw percolation on random graphs","authors":"Oliver Cooley, Abraham Guti'errez","doi":"10.4310/JOC.2020.V11.N4.A2","DOIUrl":"https://doi.org/10.4310/JOC.2020.V11.N4.A2","url":null,"abstract":"The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle. It can also be seen as a measure of whether two graphs on a common vertex set are \"jointly connected\". In this paper we consider the natural generalisation of this process to an arbitrary number of graphs on the same vertex set. We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities. This generalises a result of Bollobas, Riordan, Slivken and Smith.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"116 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77286059","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 : 2017-12-03DOI: 10.4310/JOC.2019.V10.N4.A5
Robert Davis, B. Sagan
In 2013, Beck and Braun proved and generalized multiple identities involving permutation statistics via discrete geometry. Namely, they recognized the identities as specializations of integer point transform identities for certain polyhedral cones. They extended many of their proof techniques to obtain identities involving wreath products, but some identities were resistant to their proof attempts. In this article, we provide a geometric justification of one of these wreath product identities, which was first established by Biagioli and Zeng.
{"title":"A polyhedral proof of a wreath product identity","authors":"Robert Davis, B. Sagan","doi":"10.4310/JOC.2019.V10.N4.A5","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N4.A5","url":null,"abstract":"In 2013, Beck and Braun proved and generalized multiple identities involving permutation statistics via discrete geometry. Namely, they recognized the identities as specializations of integer point transform identities for certain polyhedral cones. They extended many of their proof techniques to obtain identities involving wreath products, but some identities were resistant to their proof attempts. In this article, we provide a geometric justification of one of these wreath product identities, which was first established by Biagioli and Zeng.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"8 3","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72632908","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 : 2017-12-01DOI: 10.4310/joc.2020.v11.n4.a4
M. Ferrara, Daniel R. Johnston, Sarah Loeb, Florian Pfender, Alex Schulte, Heather C. Smith, Eric Sullivan, Michael Tait, C. Tompkins
Let $mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(mathcal{C}, t)$-saturated if $G$ does not contain any graph in $mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $mathcal{C}$. Similarly to classical saturation functions, define $mathrm{sat}_t(n, mathcal{C})$ to be the minimum number of edges in a $(mathcal{C},t)$ saturated graph. Let $mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors. In this paper we consider a variety of colored saturation problems. We determine the order of magnitude for $mathrm{sat}_t(n, mathcal{C}_r(K_k))$ for all $r$, showing a sharp change in behavior when $rgeq binom{k-1}{2}+2$. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine $mathrm{sat}_t(n, mathcal{C}_2(K_3))$ exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.
{"title":"On edge-colored saturation problems","authors":"M. Ferrara, Daniel R. Johnston, Sarah Loeb, Florian Pfender, Alex Schulte, Heather C. Smith, Eric Sullivan, Michael Tait, C. Tompkins","doi":"10.4310/joc.2020.v11.n4.a4","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n4.a4","url":null,"abstract":"Let $mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(mathcal{C}, t)$-saturated if $G$ does not contain any graph in $mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $mathcal{C}$. Similarly to classical saturation functions, define $mathrm{sat}_t(n, mathcal{C})$ to be the minimum number of edges in a $(mathcal{C},t)$ saturated graph. Let $mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors. \u0000In this paper we consider a variety of colored saturation problems. We determine the order of magnitude for $mathrm{sat}_t(n, mathcal{C}_r(K_k))$ for all $r$, showing a sharp change in behavior when $rgeq binom{k-1}{2}+2$. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine $mathrm{sat}_t(n, mathcal{C}_2(K_3))$ exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"239 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73048778","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 : 2017-11-27DOI: 10.4310/joc.2020.v11.n1.a5
R. Stanley, Fabrizio Zanello
An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and perhaps more natural, framework, by conjecturing that, for each $kge 5$, the polynomials $$f(k,m,b)(q)=binom{m}{k}_q-q^{frac{k(m-b)}{2}+b-2k+2}cdotbinom{b}{k-2}_q$$ are nonnegative and unimodal for all $mgg_k 0$ and $ble frac{km-4k+4}{k-2}$ such that $kbequiv km$ (mod 2), with the only exception of $b=frac{km-4k+2}{k-2}$ when this is an integer. Using the KOH theorem, we combinatorially show the case $k=5$. In fact, we completely characterize the nonnegativity and unimodality of $f(k,m,b)$ for $kle 5$. (This also provides an isolated counterexample to Reiner-Stanton's conjecture when $k=3$.) Further, we prove that, for each $k$ and $m$, it suffices to show our conjecture for the largest $2k-6$ values of $b$.
{"title":"A generalization of a 1998 unimodality conjecture of Reiner and Stanton","authors":"R. Stanley, Fabrizio Zanello","doi":"10.4310/joc.2020.v11.n1.a5","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n1.a5","url":null,"abstract":"An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain \"strange\" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and perhaps more natural, framework, by conjecturing that, for each $kge 5$, the polynomials $$f(k,m,b)(q)=binom{m}{k}_q-q^{frac{k(m-b)}{2}+b-2k+2}cdotbinom{b}{k-2}_q$$ are nonnegative and unimodal for all $mgg_k 0$ and $ble frac{km-4k+4}{k-2}$ such that $kbequiv km$ (mod 2), with the only exception of $b=frac{km-4k+2}{k-2}$ when this is an integer. \u0000Using the KOH theorem, we combinatorially show the case $k=5$. In fact, we completely characterize the nonnegativity and unimodality of $f(k,m,b)$ for $kle 5$. (This also provides an isolated counterexample to Reiner-Stanton's conjecture when $k=3$.) Further, we prove that, for each $k$ and $m$, it suffices to show our conjecture for the largest $2k-6$ values of $b$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"305 2 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73192524","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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