Let $M$ be an arbitrary matroid with circuits $mathcal{C}(M)$. We propose a definition of a derived matroid $delta M$ that has as its ground set $mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of $delta M$ is bounded from above by $lvert Mrvert-r(M)$ and that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the Vámos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids.
{"title":"Combinatorial Derived Matroids","authors":"Ragnar Freij, Relinde Jurrius, Olga Kuznetsova","doi":"10.37236/11327","DOIUrl":"https://doi.org/10.37236/11327","url":null,"abstract":"Let $M$ be an arbitrary matroid with circuits $mathcal{C}(M)$. We propose a definition of a derived matroid $delta M$ that has as its ground set $mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of $delta M$ is bounded from above by $lvert Mrvert-r(M)$ and that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the Vámos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"67 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87665015","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 show that a Boolean degree~$d$ function on the slice $binom{[n]}{k}$ is a junta if $k geq 2d$, and that this bound is sharp. We prove a similar result for $A$-valued degree~$d$ functions for arbitrary finite $A$, and for functions on an infinite analog of the slice.
{"title":"Junta Threshold for Low Degree Boolean Functions on the Slice","authors":"Yuval Filmus","doi":"10.37236/11115","DOIUrl":"https://doi.org/10.37236/11115","url":null,"abstract":"We show that a Boolean degree~$d$ function on the slice $binom{[n]}{k}$ is a junta if $k geq 2d$, and that this bound is sharp. We prove a similar result for $A$-valued degree~$d$ functions for arbitrary finite $A$, and for functions on an infinite analog of the slice.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"301 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136126053","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 almost Moore $(d,k)$-digraph is a regular digraph of degree $d>1$, diameter $k>1$ and order $N(d,k)=d+d^2+cdots +d^k$. So far, their existence has only been shown for $k=2$, whilst it is known that there are no such digraphs for $k=3$, $4$ and for $d=2$, $3$ when $kgeq 3$. Furthermore, under certain assumptions, the nonexistence for the remaining cases has also been shown. In this paper, we prove that $(4,k)$ and $(5,k)$-almost Moore digraphs with self-repeats do not exist for $kgeq 5$.
{"title":"Nonexistence of Almost Moore Digraphs of Degrees 4 and 5 with Self-Repeats","authors":"N. López, A. Messegué, J. Miret","doi":"10.37236/11335","DOIUrl":"https://doi.org/10.37236/11335","url":null,"abstract":"An almost Moore $(d,k)$-digraph is a regular digraph of degree $d>1$, diameter $k>1$ and order $N(d,k)=d+d^2+cdots +d^k$. So far, their existence has only been shown for $k=2$, whilst it is known that there are no such digraphs for $k=3$, $4$ and for $d=2$, $3$ when $kgeq 3$. Furthermore, under certain assumptions, the nonexistence for the remaining cases has also been shown. In this paper, we prove that $(4,k)$ and $(5,k)$-almost Moore digraphs with self-repeats do not exist for $kgeq 5$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82909343","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}
Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid $M$ with an odd circuit, if a largest odd circuit of $M$ has $k$ elements, then a largest circuit of $M$ has at most $2k-2$ elements. The goal of this note is to show that, when $M$ is $3$-connected, either $M$ has a spanning circuit, or a largest circuit of $M$ has at most $2k-4$ elements. Moreover, the latter holds when $M$ is regular of rank at least four.
Jim Geelen和Peter Nelson证明了对于一个带奇数电路的无环连接二元矩阵$M$，如果$M$的最大奇数电路有$k$个单元，则$M$的最大电路最多有$2k-2$个单元。本文的目的是说明，当$M$与$3$连接时，$M$有一个跨越电路，或者$M$的最大电路最多有$2k-4$个元件。而且，当$M$至少是秩4的正则时，后者成立。
{"title":"An Upper Bound for the Circumference of a 3-Connected Binary Matroid","authors":"Manoel Lemos, J. Oxley","doi":"10.37236/11462","DOIUrl":"https://doi.org/10.37236/11462","url":null,"abstract":"Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid $M$ with an odd circuit, if a largest odd circuit of $M$ has $k$ elements, then a largest circuit of $M$ has at most $2k-2$ elements. The goal of this note is to show that, when $M$ is $3$-connected, either $M$ has a spanning circuit, or a largest circuit of $M$ has at most $2k-4$ elements. Moreover, the latter holds when $M$ is regular of rank at least four.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"45 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80795087","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}
Antoine Dailly, Laura Eslava, A. Hansberg, Denae Ventura
Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently large, every 2-coloring of $K_n$ with more than $k$ edges in each color contains a balanced copy of $G$, then we say that $G$ is balanceable. The smallest integer $k$ such that this holds is called the balancing number of $G$.In this paper, we define a more general variant of the balancing number, the generalized balancing number, by considering 2-coverings of the edge set of $K_n$, where every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the color set ${r,b}$. In this case, edges $e$ with $L(e) = {r,b}$ act as jokers in the sense that their color can be chosen $r$ or $b$ as needed. In contrast to the balancing number, every graph has a generalized balancing number. Moreover, if the balancing number exists, then it coincides with the generalized balancing number.We give the exact value of the generalized balancing number for all cycles except for cycles of length $4k$ for which we give tight bounds. In addition, we give general bounds for the generalized balancing number of non-balanceable graphs based on the extremal number of its subgraphs, and study the generalized balancing number of $K_5$, which turns out to be surprisingly large.
{"title":"The Balancing Number and Generalized Balancing Number of Some Graph Classes","authors":"Antoine Dailly, Laura Eslava, A. Hansberg, Denae Ventura","doi":"10.37236/10032","DOIUrl":"https://doi.org/10.37236/10032","url":null,"abstract":"Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently large, every 2-coloring of $K_n$ with more than $k$ edges in each color contains a balanced copy of $G$, then we say that $G$ is balanceable. The smallest integer $k$ such that this holds is called the balancing number of $G$.In this paper, we define a more general variant of the balancing number, the generalized balancing number, by considering 2-coverings of the edge set of $K_n$, where every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the color set ${r,b}$. In this case, edges $e$ with $L(e) = {r,b}$ act as jokers in the sense that their color can be chosen $r$ or $b$ as needed. In contrast to the balancing number, every graph has a generalized balancing number. Moreover, if the balancing number exists, then it coincides with the generalized balancing number.We give the exact value of the generalized balancing number for all cycles except for cycles of length $4k$ for which we give tight bounds. In addition, we give general bounds for the generalized balancing number of non-balanceable graphs based on the extremal number of its subgraphs, and study the generalized balancing number of $K_5$, which turns out to be surprisingly large.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"97 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82748141","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}
In this article, we define a function that counts the number of (onto) homomorphisms of an oriented graph. We show that this function is always a polynomial and establish it as an extension of the notion of chromatic polynomials. We study algebraic properties of this function. In particular we show that the coefficients of these polynomials have the alternating sign property and that the polynomials associated to the independent sets have relations with the Stirling numbers of the second kind.
{"title":"A Homomorphic Polynomial for Oriented Graphs","authors":"Sandip Das, Sumitava Ghosh, S. Prabhu, Sagnik Sen","doi":"10.37236/10726","DOIUrl":"https://doi.org/10.37236/10726","url":null,"abstract":"In this article, we define a function that counts the number of (onto) homomorphisms of an oriented graph. We show that this function is always a polynomial and establish it as an extension of the notion of chromatic polynomials. We study algebraic properties of this function. In particular we show that the coefficients of these polynomials have the alternating sign property and that the polynomials associated to the independent sets have relations with the Stirling numbers of the second kind.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74632585","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}
In this paper we show that for a given $k$-tree $T$ with a $k$-clique $C$, the local mean order of all sub-$k$-trees of $T$ containing $C$ is not less than the global mean order of all sub-$k$-trees of $T$, and the path-type $k$-trees have the smallest global mean sub-$k$-tree order among all $k$-trees of a given order. These two results give solutions to two problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees. Furthermore, the mean sub-$k$-tree order as a function on $k$-trees is shown to be monotone with respect to inclusion. This generalizes Jamison's result for the case $k=1$ [J. Combin. Theory Ser. B 35 (1983), 207-223].
{"title":"On the Local and Global Mean Orders of Sub-$k$-Trees of $k$-Trees","authors":"Zuwen Luo, Kexiang Xu","doi":"10.37236/11280","DOIUrl":"https://doi.org/10.37236/11280","url":null,"abstract":"In this paper we show that for a given $k$-tree $T$ with a $k$-clique $C$, the local mean order of all sub-$k$-trees of $T$ containing $C$ is not less than the global mean order of all sub-$k$-trees of $T$, and the path-type $k$-trees have the smallest global mean sub-$k$-tree order among all $k$-trees of a given order. These two results give solutions to two problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees. Furthermore, the mean sub-$k$-tree order as a function on $k$-trees is shown to be monotone with respect to inclusion. This generalizes Jamison's result for the case $k=1$ [J. Combin. Theory Ser. B 35 (1983), 207-223].","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"175 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72511688","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. The Davenport constant $mathsf{D}(G)$ is the maximal length of minimal zero-sum sequences over $G$. For groups of the form $C_2^{r-1} oplus C_{2k}$ the Davenport constant is known for $rleq 5$. In this paper, we get the precise value of $mathsf{D}(C_2^{5} oplus C_{2k})$ for $kgeq 149$. It is also worth pointing out that our result can imply the precise value of $mathsf{D}(C_2^{4} oplus C_{2k})$.
{"title":"On Davenport Constant of the Group $C_2^{r-1} oplus C_{2k}$","authors":"K. Zhao","doi":"10.37236/11194","DOIUrl":"https://doi.org/10.37236/11194","url":null,"abstract":"Let $G$ be a finite abelian group. The Davenport constant $mathsf{D}(G)$ is the maximal length of minimal zero-sum sequences over $G$. For groups of the form $C_2^{r-1} oplus C_{2k}$ the Davenport constant is known for $rleq 5$. In this paper, we get the precise value of $mathsf{D}(C_2^{5} oplus C_{2k})$ for $kgeq 149$. It is also worth pointing out that our result can imply the precise value of $mathsf{D}(C_2^{4} oplus C_{2k})$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"29 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74083114","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 extend the classical relation between the $2n$-th Fibonacci number and the number of spanning trees of the $n$-fan graph to ribbon graphs. More importantly, we establish a relation between the $n$-associated Mersenne number and the number of quasi trees of the $n$-wheel ribbon graph. The calculations are performed by computing the determinant of a matrix associated with ribbon graphs. These theorems are also proven using contraction and deletion in ribbon graphs. The results provide neat and symmetric combinatorial interpretations of these well-known sequences. Furthermore, they are refined by giving two families of abelian groups whose orders are the Fibonacci and associated Mersenne numbers.
{"title":"The Number of Quasi-Trees in Fans and Wheels","authors":"C. Merino","doi":"10.37236/11097","DOIUrl":"https://doi.org/10.37236/11097","url":null,"abstract":"We extend the classical relation between the $2n$-th Fibonacci number and the number of spanning trees of the $n$-fan graph to ribbon graphs. More importantly, we establish a relation between the $n$-associated Mersenne number and the number of quasi trees of the $n$-wheel ribbon graph. The calculations are performed by computing the determinant of a matrix associated with ribbon graphs. These theorems are also proven using contraction and deletion in ribbon graphs. The results provide neat and symmetric combinatorial interpretations of these well-known sequences. Furthermore, they are refined by giving two families of abelian groups whose orders are the Fibonacci and associated Mersenne numbers.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"22 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83386884","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 Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to lie in one of several classes, and obtain strong new constraints on their mutual relationships. An important consideration in determining these classes is whether the strongly regular graph associated with the balanced splittable Hadamard matrix is primitive or imprimitive. We construct new infinite families of balanced splittable Hadamard matrices in both the primitive and imprimitive cases. A rich source of examples is provided by packings of partial difference sets in elementary abelian $2$-groups, from which we construct Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition.
{"title":"Constructions and Restrictions for Balanced Splittable Hadamard Matrices","authors":"Jonathan Jedwab, Shuxing Li, Samuel Simon","doi":"10.37236/11586","DOIUrl":"https://doi.org/10.37236/11586","url":null,"abstract":"A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to lie in one of several classes, and obtain strong new constraints on their mutual relationships. An important consideration in determining these classes is whether the strongly regular graph associated with the balanced splittable Hadamard matrix is primitive or imprimitive. We construct new infinite families of balanced splittable Hadamard matrices in both the primitive and imprimitive cases. A rich source of examples is provided by packings of partial difference sets in elementary abelian $2$-groups, from which we construct Hadamard matrices admitting a row decomposition so that the balanced splittable property holds simultaneously with respect to every union of the submatrices of the decomposition.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136166513","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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