Susan Jowett, Jasmine Lulani Kaulamatoa, G. Whittle
If $(X,Y)$ is a partition of the vertices of a graph $G=(V,E)$ and there are $k$ edges joining vertices in $X$ to vertices in $Y$, then $(X,Y)$ is an edge separation of $G$ of order $k$. The graph $G$ is $(n,k)$-edge connected, if whenever $(X,Y)$ is an edge separation of $G$ of order at most $k$, then either $X$ or $Y$ has at most $n$ elements. We prove that if $G$ is cubic and $(n,k)$-edge connected, then one can find edges to delete so that the resulting graph is $(6n+2,k)$-edge connected. We find an explicit bound on the size of a cubic graph that is minimal in the immersion order with respect to having carving-width $k$. The techniques we use generalise techniques used to prove similar theorems for other structures. In an attempt to develop a unified setting we set up an axiomatic framework to describe certain classes of connectivity functions. We prove a theorem for such classes that gives sufficient conditions to enable a bound on the size of members that are minimal with respect to having branch-width greater than $k$. As well as proving the above mentioned result for edge connectivity in this setting, we prove (known) bounds on the size of excluded minors for the classes of matroids and graphs of branch-width $k$. We also bound the size of a connectivity function that has branch-width greater than $k$ and is minimal with respect to an operation known as elision.
{"title":"Bounding Branch-Width","authors":"Susan Jowett, Jasmine Lulani Kaulamatoa, G. Whittle","doi":"10.37236/11162","DOIUrl":"https://doi.org/10.37236/11162","url":null,"abstract":"If $(X,Y)$ is a partition of the vertices of a graph $G=(V,E)$ and there are $k$ edges joining vertices in $X$ to vertices in $Y$, then $(X,Y)$ is an edge separation of $G$ of order $k$. The graph $G$ is $(n,k)$-edge connected, if whenever $(X,Y)$ is an edge separation of $G$ of order at most $k$, then either $X$ or $Y$ has at most $n$ elements. We prove that if $G$ is cubic and $(n,k)$-edge connected, then one can find edges to delete so that the resulting graph is $(6n+2,k)$-edge connected. We find an explicit bound on the size of a cubic graph that is minimal in the immersion order with respect to having carving-width $k$. The techniques we use generalise techniques used to prove similar theorems for other structures. In an attempt to develop a unified setting we set up an axiomatic framework to describe certain classes of connectivity functions. We prove a theorem for such classes that gives sufficient conditions to enable a bound on the size of members that are minimal with respect to having branch-width greater than $k$. As well as proving the above mentioned result for edge connectivity in this setting, we prove (known) bounds on the size of excluded minors for the classes of matroids and graphs of branch-width $k$. We also bound the size of a connectivity function that has branch-width greater than $k$ and is minimal with respect to an operation known as elision.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"2 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83653603","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 balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the complete graph on $n=13^k$ vertices, for every positive integer $k$, with no rainbow $K_4$. This solves a problem by Erdős and Tuza.
{"title":"Balanced Edge-Colorings Avoiding Rainbow Cliques of Size Four","authors":"F. Clemen, Adam Zsolt Wagner","doi":"10.37236/11965","DOIUrl":"https://doi.org/10.37236/11965","url":null,"abstract":"A balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the complete graph on $n=13^k$ vertices, for every positive integer $k$, with no rainbow $K_4$. This solves a problem by Erdős and Tuza.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73739104","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}
An inversion sequence of length $n$ is a sequence of integers $e=e_1cdots e_n$ which satisfies for each $iin[n]={1,2,ldots,n}$ the inequality $0le e_i < i$. For a set of patterns $P$, we let $mathbf{I}_n(P)$ denote the set of inversion sequences of length $n$ that avoid all the patterns from~$P$. We say that two sets of patterns $P$ and $Q$ are I-Wilf-equivalent if $|mathbf{I}_n(P)|=|mathbf{I}_n(Q)|$ for every~$n$. In this paper, we show that the number of I-Wilf-equivalence classes among triples of length-3 patterns is $137$, $138$ or~$139$. In particular, to show that this number is exactly $137$, it remains to prove ${101,102,110}stackrel{mathbf{I}}{sim}{021,100,101}$ and ${100,110,201}stackrel{mathbf{I}}{sim}{100,120,210}$.
长度为$n$的反转序列是一个整数序列$e=e_1cdots e_n$，它满足每个$iin[n]={1,2,ldots,n}$不等式$0le e_i < i$。对于一组模式$P$，我们让$mathbf{I}_n(P)$表示一组长度为$n$的反转序列，这些序列避免了来自$P$的所有模式。我们说，如果$|mathbf{I}_n(P)|=|mathbf{I}_n(Q)|$对应每个$n$，那么两组模式$P$和$Q$是等价的。在本文中，我们证明了长度为3模式的三元组中i - will -等价类的个数为$137$, $138$或$139$。特别是，为了证明这个数字确实是$137$，还需要证明${101,102,110}stackrel{mathbf{I}}{sim}{021,100,101}$和${100,110,201}stackrel{mathbf{I}}{sim}{100,120,210}$。
{"title":"Inversion Sequences Avoiding a Triple of Patterns of 3 Letters","authors":"David Callan, Vít Jelínek, T. Mansour","doi":"10.37236/11603","DOIUrl":"https://doi.org/10.37236/11603","url":null,"abstract":"An inversion sequence of length $n$ is a sequence of integers $e=e_1cdots e_n$ which satisfies for each $iin[n]={1,2,ldots,n}$ the inequality $0le e_i < i$. For a set of patterns $P$, we let $mathbf{I}_n(P)$ denote the set of inversion sequences of length $n$ that avoid all the patterns from~$P$. We say that two sets of patterns $P$ and $Q$ are I-Wilf-equivalent if $|mathbf{I}_n(P)|=|mathbf{I}_n(Q)|$ for every~$n$. In this paper, we show that the number of I-Wilf-equivalence classes among triples of length-3 patterns is $137$, $138$ or~$139$. In particular, to show that this number is exactly $137$, it remains to prove ${101,102,110}stackrel{mathbf{I}}{sim}{021,100,101}$ and ${100,110,201}stackrel{mathbf{I}}{sim}{100,120,210}$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79272710","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}
Nicolas Bousquet, Bastien Durain, None Théo Pierron, Stéphan Thomassé
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.
{"title":"Extremal Independent Set Reconfiguration","authors":"Nicolas Bousquet, Bastien Durain, None Théo Pierron, Stéphan Thomassé","doi":"10.37236/11771","DOIUrl":"https://doi.org/10.37236/11771","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":"594 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135556305","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}
Siddharth Berera, Andrés Gómez-Colunga, Joey Lakerdas-Gayle, John López, Mauditra Matin, Daniel Roebuck, Eric Rowland, Noam Scully, Juliet Whidden
The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$. When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the ruler sequence. For other prefixes, the structure is significantly more complicated. In this paper, we show that $L(p)$ reflects the structure of the ruler sequence for several words $p$. We provide morphisms that generate $L(n)$ for letters $n=1$ and $ngeq3$, and $L(p)$ for most families of two-letter words $p$.
{"title":"The Lexicographically Least Square-Free Word with a Given Prefix","authors":"Siddharth Berera, Andrés Gómez-Colunga, Joey Lakerdas-Gayle, John López, Mauditra Matin, Daniel Roebuck, Eric Rowland, Noam Scully, Juliet Whidden","doi":"10.37236/11659","DOIUrl":"https://doi.org/10.37236/11659","url":null,"abstract":"The lexicographically least square-free infinite word on the alphabet of non-negative integers with a given prefix $p$ is denoted $L(p)$. When $p$ is the empty word, this word was shown by Guay-Paquet and Shallit to be the ruler sequence. For other prefixes, the structure is significantly more complicated. In this paper, we show that $L(p)$ reflects the structure of the ruler sequence for several words $p$. We provide morphisms that generate $L(n)$ for letters $n=1$ and $ngeq3$, and $L(p)$ for most families of two-letter words $p$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135556306","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}
Four problems about tilings, related to the so-called: Heesch number, isohedral number, $m$-morphic figures, and $sigma$-morphic figures, can be asked in four variations of the notion of tiling: protosets with more elements, disconnected tiles, colored tiles and tessellations in larger-dimensional spaces. That makes $16$ combinations in total. Five among them have been previously solved in the literature, and one has been partially solved. We here solve seven of the remaining combinations, and additionally complete that partial solution.
{"title":"Solutions to Seven and a Half Problems on Tilings","authors":"Bojan Bašić, Aleksa Džuklevski, Anna Slivková","doi":"10.37236/11813","DOIUrl":"https://doi.org/10.37236/11813","url":null,"abstract":"Four problems about tilings, related to the so-called: Heesch number, isohedral number, $m$-morphic figures, and $sigma$-morphic figures, can be asked in four variations of the notion of tiling: protosets with more elements, disconnected tiles, colored tiles and tessellations in larger-dimensional spaces. That makes $16$ combinations in total. Five among them have been previously solved in the literature, and one has been partially solved. We here solve seven of the remaining combinations, and additionally complete that partial solution.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"5 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74288696","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 $n$ be the order of a quaternary Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable quaternary Hadamard matrix of order $n^2$.
{"title":"Hadamard Matrices related to Projective Planes","authors":"Hadi Kharaghani, Sho Suda","doi":"10.37236/11990","DOIUrl":"https://doi.org/10.37236/11990","url":null,"abstract":"Let $n$ be the order of a quaternary Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable quaternary Hadamard matrix of order $n^2$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"182 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136064949","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}
Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur polynomials. We prove a nonsymmetric generalization of Monk's rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a degree one key polynomial.
{"title":"Monk's Rule for Demazure Characters of the General Linear Group","authors":"Sami H. Assaf, Danjoseph Quijada","doi":"10.37236/11425","DOIUrl":"https://doi.org/10.37236/11425","url":null,"abstract":"Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur polynomials. We prove a nonsymmetric generalization of Monk's rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a degree one key polynomial.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"46 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73811506","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 hypergraph is properly vertex-colored if no edge contains vertices which are assigned the same color. We provide an algebraic formulation of the $k$-colorability of uniform and non-uniform hypergraphs. This formulation provides an algebraic algorithm, via Gröbner bases, which can determine whether a given hypergraph is $k$-colorable or not. We further study new families of k-colorings with additional restrictions on permissible colorings. These new families of colorings generalize several recently studied variations of $k$-colorings.
{"title":"An Algebraic Formulation of Hypergraph Colorings","authors":"Michael Krul, L. Thoma","doi":"10.37236/9894","DOIUrl":"https://doi.org/10.37236/9894","url":null,"abstract":"A hypergraph is properly vertex-colored if no edge contains vertices which are assigned the same color. We provide an algebraic formulation of the $k$-colorability of uniform and non-uniform hypergraphs. This formulation provides an algebraic algorithm, via Gröbner bases, which can determine whether a given hypergraph is $k$-colorable or not. We further study new families of k-colorings with additional restrictions on permissible colorings. These new families of colorings generalize several recently studied variations of $k$-colorings.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"291 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74852573","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}
Dariusz Dereniowski, Przemysław Gordinowicz, Paweł Prałat
We investigate two types of query games played on a graph, pair queries and edge queries. We concentrate on investigating the two associated graph parameters for binomial random graphs, and showing that determining any of the two parameters is NP-hard for bounded degree graphs.
{"title":"Edge and Pair Queries-Random Graphs and Complexity","authors":"Dariusz Dereniowski, Przemysław Gordinowicz, Paweł Prałat","doi":"10.37236/11159","DOIUrl":"https://doi.org/10.37236/11159","url":null,"abstract":"We investigate two types of query games played on a graph, pair queries and edge queries. We concentrate on investigating the two associated graph parameters for binomial random graphs, and showing that determining any of the two parameters is NP-hard for bounded degree graphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136039542","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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