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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Pub Date : 2021-10-25DOI: 10.26493/1855-3974.2805.b49
J'anos Bar'at, G'eza T'oth
A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is, maximal plane graphs, of $n$ vertices have exactly $3n-6$ edges. For $k>0$, the number of edges of saturated $n$-vertex $k$-plane graphs can take many different values. In this note, we establish some bounds on the minimum number of edges of saturated $2$-plane graphs under different conditions. If two edges can cross at most once, then such a graph has at least $n-1$ edges. If two edges can cross many times, then we show the tight bound of $lfloor2n/3rfloor$ for the number of edges.
{"title":"Saturated 2-plane drawings with few edges","authors":"J'anos Bar'at, G'eza T'oth","doi":"10.26493/1855-3974.2805.b49","DOIUrl":"https://doi.org/10.26493/1855-3974.2805.b49","url":null,"abstract":"A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is, maximal plane graphs, of $n$ vertices have exactly $3n-6$ edges. For $k>0$, the number of edges of saturated $n$-vertex $k$-plane graphs can take many different values. In this note, we establish some bounds on the minimum number of edges of saturated $2$-plane graphs under different conditions. If two edges can cross at most once, then such a graph has at least $n-1$ edges. If two edges can cross many times, then we show the tight bound of $lfloor2n/3rfloor$ for the number of edges.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87245514","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 : 2021-10-05DOI: 10.26493/1855-3974.2710.f3d
R. Burdett, M. Haythorpe, Alex Newcombe
We consider the flower snarks, a widely studied infinite family of 3--regular graphs. For the Flower snark $J_n$ on $4n$ vertices, it is trivial to show that the domination number of $J_n$ is equal to $n$. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure Domination and weak Roman domination numbers for flower snarks.
{"title":"Variants of the domination number for flower snarks","authors":"R. Burdett, M. Haythorpe, Alex Newcombe","doi":"10.26493/1855-3974.2710.f3d","DOIUrl":"https://doi.org/10.26493/1855-3974.2710.f3d","url":null,"abstract":"We consider the flower snarks, a widely studied infinite family of 3--regular graphs. For the Flower snark $J_n$ on $4n$ vertices, it is trivial to show that the domination number of $J_n$ is equal to $n$. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure Domination and weak Roman domination numbers for flower snarks.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76839834","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 : 2021-09-07DOI: 10.26493/1855-3974.2701.b7d
H. Prodinger
Following an orginal idea by Kn"odel, an online bin-packing problem is considered where the the large items arrive in double-packs. The dual problem where the small items arrive in double-packs is also considered. The enumerations have a ternary random walk flavour, and for the enumeration, the kernel method is employed.
{"title":"An online bin-packing problem with an underlying ternary structure","authors":"H. Prodinger","doi":"10.26493/1855-3974.2701.b7d","DOIUrl":"https://doi.org/10.26493/1855-3974.2701.b7d","url":null,"abstract":"Following an orginal idea by Kn\"odel, an online bin-packing problem is considered where the the large items arrive in double-packs. The dual problem where the small items arrive in double-packs is also considered. The enumerations have a ternary random walk flavour, and for the enumeration, the kernel method is employed.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78159527","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 : 2021-07-14DOI: 10.26493/1855-3974.2329.97A
Z. Stanić
It is known that a signed graph with exactly 2 eigenvalues must be regular, and all those whose vertex degree does not exceed 4 are known. In this paper we characterize all signed graphs with 2 eigenvalues and vertex degree 5. We also determine all signed graphs with 2 eigenvalues and 12 or 13 vertices, which is a natural step since those with a fewer number of vertices are known.
{"title":"Signed graphs with two eigenvalues and vertex degree five","authors":"Z. Stanić","doi":"10.26493/1855-3974.2329.97A","DOIUrl":"https://doi.org/10.26493/1855-3974.2329.97A","url":null,"abstract":"It is known that a signed graph with exactly 2 eigenvalues must be regular, and all those whose vertex degree does not exceed 4 are known. In this paper we characterize all signed graphs with 2 eigenvalues and vertex degree 5. We also determine all signed graphs with 2 eigenvalues and 12 or 13 vertices, which is a natural step since those with a fewer number of vertices are known.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75493421","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 : 2021-06-12DOI: 10.26493/1855-3974.3094.bc6
J. Tuite, Elias John Thomas, Ullas Chandran S.V.
The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers, with applications to a realisation result. We then solve a Turan problem for the size of graphs with given order and position numbers and characterise the possible diameters of graphs with given order and monophonic position number. Finally we classify the graphs with given order and diameter and largest possible general position number.
{"title":"On some extremal position problems for graphs","authors":"J. Tuite, Elias John Thomas, Ullas Chandran S.V.","doi":"10.26493/1855-3974.3094.bc6","DOIUrl":"https://doi.org/10.26493/1855-3974.3094.bc6","url":null,"abstract":"The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers, with applications to a realisation result. We then solve a Turan problem for the size of graphs with given order and position numbers and characterise the possible diameters of graphs with given order and monophonic position number. Finally we classify the graphs with given order and diameter and largest possible general position number.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81287842","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 : 2021-05-27DOI: 10.26493/1855-3974.2688.2de
Jingnan Xie, Yan-Quan Feng, Jin-Xin Zhou
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $alphain Aut(G)$ such that $S^alpha=T$, whenever $Cay(G,S)cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture proposed by 'Ad'am in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It takes about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order $2n$ is a NCI-group or a NDCI-group if and only if $n=2,4$ or $n$ is odd. As a direct consequence, we have that if a dihedral group $D_{2n}$ of order $2n$ is a DCI-group then $n=2$ or $n$ is odd-square-free, and that if $D_{2n}$ is a CI-group then $n=2,9$ or $n$ is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups.
{"title":"Normal Cayley digraphs of dihedral groups with CI-property","authors":"Jingnan Xie, Yan-Quan Feng, Jin-Xin Zhou","doi":"10.26493/1855-3974.2688.2de","DOIUrl":"https://doi.org/10.26493/1855-3974.2688.2de","url":null,"abstract":"A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $alphain Aut(G)$ such that $S^alpha=T$, whenever $Cay(G,S)cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture proposed by 'Ad'am in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It takes about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order $2n$ is a NCI-group or a NDCI-group if and only if $n=2,4$ or $n$ is odd. As a direct consequence, we have that if a dihedral group $D_{2n}$ of order $2n$ is a DCI-group then $n=2$ or $n$ is odd-square-free, and that if $D_{2n}$ is a CI-group then $n=2,9$ or $n$ is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80394947","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 : 2021-03-22DOI: 10.26493/1855-3974.2616.4a9
Simone Costa, T. Traetta
Let F = {Fα : α ∈ A} be a family of infinite graphs, together with Λ. The Factorization Problem FP (F ,Λ) asks whether F can be realized as a factorization of Λ, namely, whether there is a factorization G = {Γα : α ∈ A} of Λ such that each Γα is a copy of Fα. We study this problem when Λ is either the Rado graph R or the complete graph Kא of infinite order א. When F is a countable family, we show that FP (F , R) is solvable if and only if each graph in F has no finite dominating set. We also prove that FP (F ,Kא) admits a solution whenever the cardinality F coincide with the order and the domination numbers of its graphs. For countable complete graphs, we show some non existence results when the domination numbers of the graphs in F are finite. More precisely, we show that there is no factorization of KN into copies of a k-star (that is, the vertex disjoint union of k countable stars) when k = 1, 2, whereas it exists when k ≥ 4, leaving the problem open for k = 3. Finally, we determine sufficient conditions for the graphs of a decomposition to be arranged into resolution classes.
{"title":"Factorizing the Rado graph and infinite complete graphs","authors":"Simone Costa, T. Traetta","doi":"10.26493/1855-3974.2616.4a9","DOIUrl":"https://doi.org/10.26493/1855-3974.2616.4a9","url":null,"abstract":"Let F = {Fα : α ∈ A} be a family of infinite graphs, together with Λ. The Factorization Problem FP (F ,Λ) asks whether F can be realized as a factorization of Λ, namely, whether there is a factorization G = {Γα : α ∈ A} of Λ such that each Γα is a copy of Fα. We study this problem when Λ is either the Rado graph R or the complete graph Kא of infinite order א. When F is a countable family, we show that FP (F , R) is solvable if and only if each graph in F has no finite dominating set. We also prove that FP (F ,Kא) admits a solution whenever the cardinality F coincide with the order and the domination numbers of its graphs. For countable complete graphs, we show some non existence results when the domination numbers of the graphs in F are finite. More precisely, we show that there is no factorization of KN into copies of a k-star (that is, the vertex disjoint union of k countable stars) when k = 1, 2, whereas it exists when k ≥ 4, leaving the problem open for k = 3. Finally, we determine sufficient conditions for the graphs of a decomposition to be arranged into resolution classes.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79182728","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：