Pub Date : 2023-02-13DOI: 10.26493/1855-3974.2800.d12
Qi'an Chen, Siddhant Jajodia, T. Jordán, Kate Perkins
By mapping the vertices of a graph G to points in R 3 , and its edges to the corresponding line segments, we obtain a three-dimensional realization of G . A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3 n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.
{"title":"Redundantly globally rigid braced triangulations","authors":"Qi'an Chen, Siddhant Jajodia, T. Jordán, Kate Perkins","doi":"10.26493/1855-3974.2800.d12","DOIUrl":"https://doi.org/10.26493/1855-3974.2800.d12","url":null,"abstract":"By mapping the vertices of a graph G to points in R 3 , and its edges to the corresponding line segments, we obtain a three-dimensional realization of G . A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3 n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83327042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-13DOI: 10.26493/1855-3974.2976.f76
Pavle Saksida
{"title":"On the beta distribution, the nonlinear Fourier transform and a combinatorial problem","authors":"Pavle Saksida","doi":"10.26493/1855-3974.2976.f76","DOIUrl":"https://doi.org/10.26493/1855-3974.2976.f76","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"64 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76516324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-19DOI: 10.26493/1855-3974.3071.37e
Dario Sterzi, Pablo Spiga
Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.
{"title":"Almost all Cayley maps are mapical regular representations","authors":"Dario Sterzi, Pablo Spiga","doi":"10.26493/1855-3974.3071.37e","DOIUrl":"https://doi.org/10.26493/1855-3974.3071.37e","url":null,"abstract":"Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78596508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-06DOI: 10.26493/1855-3974.2672.73b
Marién Abreu, John Baptist Gauci, Domenico Labbate, Federico Romaniello, Jean Paul Zerafa
A graph G has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of G such that the union of the two perfect matchings yields a Hamiltonian cycle of G. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if G is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic.
{"title":"Perfect matchings, Hamiltonian cycles and edge-colourings in a class of cubic graphs","authors":"Marién Abreu, John Baptist Gauci, Domenico Labbate, Federico Romaniello, Jean Paul Zerafa","doi":"10.26493/1855-3974.2672.73b","DOIUrl":"https://doi.org/10.26493/1855-3974.2672.73b","url":null,"abstract":"A graph G has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of G such that the union of the two perfect matchings yields a Hamiltonian cycle of G. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if G is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135223597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-05DOI: 10.26493/1855-3974.2975.1b2
Luca Sabatini
Let $G$ be a group. We give an explicit description of the set of elements $x in G$ such that $x^{|G:H|} in H$ for every subgroup of finite index $H leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?
让$G$成为一个团体。我们给出了元素集合$x in G$的显式描述，使得$x^{|G:H|} in H$对于有限索引$H leqslant G$的每一子群。这涉及到以下问题:给定两个子群$H$和$K$, $H$的索引是有限的，$|HK:H|$何时能除$|G:H|$ ?
{"title":"Products of subgroups, subnormality, and relative orders of elements","authors":"Luca Sabatini","doi":"10.26493/1855-3974.2975.1b2","DOIUrl":"https://doi.org/10.26493/1855-3974.2975.1b2","url":null,"abstract":"Let $G$ be a group. We give an explicit description of the set of elements $x in G$ such that $x^{|G:H|} in H$ for every subgroup of finite index $H leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"42 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78610886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-14DOI: 10.26493/1855-3974.2894.b07
John H. Johnson, Max Wakefield
There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M"obius function. Under the product this generalized M"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M"obius functions. Using this generalized M"obius function we define analogues of the characteristic polynomial and M"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.
{"title":"A non-associative incidence near-ring with a generalized Möbius function","authors":"John H. Johnson, Max Wakefield","doi":"10.26493/1855-3974.2894.b07","DOIUrl":"https://doi.org/10.26493/1855-3974.2894.b07","url":null,"abstract":"There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M\"obius function. Under the product this generalized M\"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M\"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M\"obius functions. Using this generalized M\"obius function we define analogues of the characteristic polynomial and M\"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M\"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"32 Suppl 3 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87136666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-22DOI: 10.26493/1855-3974.2947.cd6
Martin Bachrat'y
A skew morphism of a finite group $B$ is a permutation $varphi$ of $B$ that preserves the identity element of $B$ and has the property that for every $ain B$ there exists a positive integer $i_a$ such that $varphi(ab) = varphi(a)varphi^{i_a}(b)$ for all $bin B$. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism $varphi$ of $mathbb{Z}_n$ is closely related to a specific skew morphism of $mathbb{Z}_{|!langle varphi rangle!|}$, called the quotient of $varphi$. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order $2^em$ with $ein {0,1,2,3,4}$ and $m$ odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to $161$. During the preparation of this paper we noticed a few flaws in Section~5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68-100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.
有限群$B$的偏态射是$B$的一个排列$varphi$，它保留了$B$的单位元，并且具有这样的性质:对于B$中的每一个$ A ，存在一个正整数$i_a$，使得$varphi(ab) = varphi(A)varphi^{i_a}(B)$对于B$中的所有$B 。对所有有限循环群的斜态射进行分类是出了名的困难，目前还没有这样的分类。$mathbb{Z}_n$的每个偏态$varphi$与$mathbb{Z}_{|!langle varphi rangle!|}$，称为$varphi$的商。本文利用这一关系和其他观察结果，证明了有限循环群的斜态射的一些新定理。特别地，我们对所有$2^em$阶循环群的偏态进行了分类，其中$e In {0,1,2,3,4}$和$m$为奇数和无平方。我们还开发了一种寻找环群的偏态射的算法，并在MAGMA中实现了该算法，得到了阶为$161$的环群的所有偏态射的普查。在本文的准备过程中，我们注意到论文环补和群的偏态射的第~5节有一些缺陷[J]。代数[j].， 2016，(6)， 68-100。我们提出并证明了有问题的原始断言的弱版本(即引理5.3(b)，定理5.6和推论5.7)，并表明我们的修改可以用于修复使用这些有问题断言中的至少一个的所有结果证明(在上述论文中)。
{"title":"Quotients of skew morphisms of cyclic groups","authors":"Martin Bachrat'y","doi":"10.26493/1855-3974.2947.cd6","DOIUrl":"https://doi.org/10.26493/1855-3974.2947.cd6","url":null,"abstract":"A skew morphism of a finite group $B$ is a permutation $varphi$ of $B$ that preserves the identity element of $B$ and has the property that for every $ain B$ there exists a positive integer $i_a$ such that $varphi(ab) = varphi(a)varphi^{i_a}(b)$ for all $bin B$. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism $varphi$ of $mathbb{Z}_n$ is closely related to a specific skew morphism of $mathbb{Z}_{|!langle varphi rangle!|}$, called the quotient of $varphi$. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order $2^em$ with $ein {0,1,2,3,4}$ and $m$ odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to $161$. During the preparation of this paper we noticed a few flaws in Section~5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68-100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82509403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-20DOI: 10.26493/1855-3974.2903.9ca
A. Aguglia, Bence Csajb'ok, Zsuzsa Weiner
Let $U$ be a set of polynomials of degree at most $k$ over $mathbb{F}_q$, the finite field of $q$ elements. Assume that $U$ is an intersecting family, that is, the graphs of any two of the polynomials in $U$ share a common point. Adriaensen proved that the size of $U$ is at most $q^k$ with equality if and only if $U$ is the set of all polynomials of degree at most $k$ passing through a common point. In this manuscript, using a different, polynomial approach, we prove a stability version of this result, that is, the same conclusion holds if $|U|>q^k-q^{k-1}$. We prove a stronger result when $k=2$. For our purposes, we also prove the following results. If the set of directions determined by the graph of $f$ is contained in an additive subgroup of $mathbb{F}_q$, then the graph of $f$ is a line. If the set of directions determined by at least $q-sqrt{q}/2$ affine points is contained in the set of squares/non-squares plus the common point of either the vertical or the horizontal lines, then up to an affinity the point set is contained in the graph of some polynomial of the form $alpha x^{p^k}$.
{"title":"Intersecting families of graphs of functions over a finite field","authors":"A. Aguglia, Bence Csajb'ok, Zsuzsa Weiner","doi":"10.26493/1855-3974.2903.9ca","DOIUrl":"https://doi.org/10.26493/1855-3974.2903.9ca","url":null,"abstract":"Let $U$ be a set of polynomials of degree at most $k$ over $mathbb{F}_q$, the finite field of $q$ elements. Assume that $U$ is an intersecting family, that is, the graphs of any two of the polynomials in $U$ share a common point. Adriaensen proved that the size of $U$ is at most $q^k$ with equality if and only if $U$ is the set of all polynomials of degree at most $k$ passing through a common point. In this manuscript, using a different, polynomial approach, we prove a stability version of this result, that is, the same conclusion holds if $|U|>q^k-q^{k-1}$. We prove a stronger result when $k=2$. For our purposes, we also prove the following results. If the set of directions determined by the graph of $f$ is contained in an additive subgroup of $mathbb{F}_q$, then the graph of $f$ is a line. If the set of directions determined by at least $q-sqrt{q}/2$ affine points is contained in the set of squares/non-squares plus the common point of either the vertical or the horizontal lines, then up to an affinity the point set is contained in the graph of some polynomial of the form $alpha x^{p^k}$.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"519 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72418970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-20DOI: 10.26493/1855-3974.2826.3dc
Sean Dewar, J. Hewetson, A. Nixon
A bar-joint framework $(G,p)$ is the combination of a graph $G$ and a map $p$ assigning positions, in some space, to the vertices of $G$. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.
{"title":"Coincident-point rigidity in normed planes","authors":"Sean Dewar, J. Hewetson, A. Nixon","doi":"10.26493/1855-3974.2826.3dc","DOIUrl":"https://doi.org/10.26493/1855-3974.2826.3dc","url":null,"abstract":"A bar-joint framework $(G,p)$ is the combination of a graph $G$ and a map $p$ assigning positions, in some space, to the vertices of $G$. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"81 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88449633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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