Pub Date : 2020-11-30DOI: 10.26493/1855-3974.2863.4b9
Marcin Stawiski
Call a colouring of a graph emph{distinguishing} if the only automorphism which preserves it is the identity. We investigate the role of the Axiom of Choice in the existence of certain proper or distinguishing colourings in both vertex and edge variants with emphasis on locally finite connected graphs. In particular, we show that every locally finite connected graph has a distinguishing or proper colouring if and only if KH{o}nig's Lemma holds. We show that we cannot prove in ZF that such colourings exist even for connected graphs with maximum degree 3. We also formulate few conditions about distinguishing and proper colouring which are equivalent to the Axiom of Choice.
{"title":"The role of the Axiom of Choice in proper and distinguishing colourings","authors":"Marcin Stawiski","doi":"10.26493/1855-3974.2863.4b9","DOIUrl":"https://doi.org/10.26493/1855-3974.2863.4b9","url":null,"abstract":"Call a colouring of a graph emph{distinguishing} if the only automorphism which preserves it is the identity. We investigate the role of the Axiom of Choice in the existence of certain proper or distinguishing colourings in both vertex and edge variants with emphasis on locally finite connected graphs. In particular, we show that every locally finite connected graph has a distinguishing or proper colouring if and only if KH{o}nig's Lemma holds. We show that we cannot prove in ZF that such colourings exist even for connected graphs with maximum degree 3. We also formulate few conditions about distinguishing and proper colouring which are equivalent to the Axiom of Choice.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78633051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-22DOI: 10.26493/1855-3974.1851.b44
H. Qu, Yan Wang, Kai Yuan
In this paper, we prove that a Frobenius group (except for those which are dihedral groups) can only be the automorphism group of an orientably-regular chiral map. The necessary and sufficient conditions for a Frobenius group to be the automorphism group of an orientably-regular chiral map are given. Furthermore, these orientably-regular chiral maps with Frobenius automorphisms are proved to be normal Cayley maps. Frobenius groups conforming to these conditions are also constructed.
{"title":"Frobenius groups which are the automorphism groups of orientably-regular maps","authors":"H. Qu, Yan Wang, Kai Yuan","doi":"10.26493/1855-3974.1851.b44","DOIUrl":"https://doi.org/10.26493/1855-3974.1851.b44","url":null,"abstract":"In this paper, we prove that a Frobenius group (except for those which are dihedral groups) can only be the automorphism group of an orientably-regular chiral map. The necessary and sufficient conditions for a Frobenius group to be the automorphism group of an orientably-regular chiral map are given. Furthermore, these orientably-regular chiral maps with Frobenius automorphisms are proved to be normal Cayley maps. Frobenius groups conforming to these conditions are also constructed.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84488177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-21DOI: 10.26493/1855-3974.2262.9B8
Wuxing Cai, Yuehui Chen, Lili Mu
Let A be a hyperplane arrangement in A isomorphic to ℝ n . Let V q be the q -Varchenko matrix for the arrangement A with all hyperplane parameters equal to q . In this paper, we consider three interesting cases of q -Varchenko matrices associated to hyperplane arrangements. We show that they have a Smith normal form over ℤ[ q ] .
{"title":"On the Smith normal form of Varchenko matrix","authors":"Wuxing Cai, Yuehui Chen, Lili Mu","doi":"10.26493/1855-3974.2262.9B8","DOIUrl":"https://doi.org/10.26493/1855-3974.2262.9B8","url":null,"abstract":"Let A be a hyperplane arrangement in A isomorphic to ℝ n . Let V q be the q -Varchenko matrix for the arrangement A with all hyperplane parameters equal to q . In this paper, we consider three interesting cases of q -Varchenko matrices associated to hyperplane arrangements. We show that they have a Smith normal form over ℤ[ q ] .","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74864627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-20DOI: 10.26493/1855-3974.2122.1E2
Xue Wang, F. Yin, Jin-Xin Zhou
For a k -regular graph Γ and a graph Υ of order k , a generalized truncation of Γ by Υ is constructed by replacing each vertex of Γ with a copy of Υ . E. Eiben, R. Jajcay and P. S parl introduced a method for constructing vertex-transitive generalized truncations. For convenience, we call a graph obtained by using Eiben et al. ’s method a special generalized truncation . In their paper, Eiben et al. proposed a problem to classify special generalized truncations of a complete graph K n by a cycle of length n − 1 . In this paper, we completely solve this problem by demonstrating that with the exception of n = 6 , every special generalized truncation of a complete graph K n by a cycle of length n − 1 is a Cayley graph of AGL(1, n ) where n is a prime power. Moreover, the full automorphism groups of all these graphs and the isomorphisms among them are determined.
对于k正则图Γ和k阶图Υ，通过将Γ的每个顶点替换为Υ的副本来构造Γ被Υ的广义截断。E. Eiben, R. Jajcay和P. S . parl介绍了一种构造顶点传递广义截断的方法。为方便起见，我们称用Eiben等人的方法得到的图为特殊广义截断。Eiben等人在他们的论文中提出了一个用长度为n−1的循环对完全图K n的特殊广义截断进行分类的问题。在本文中，我们证明了除n = 6外，完全图K n被一个长度为n−1的环截断的任何特殊广义截断都是AGL(1, n)的Cayley图，其中n是一个素数幂。并确定了所有图的全自同构群及其同构。
{"title":"On generalized truncations of complete graphs","authors":"Xue Wang, F. Yin, Jin-Xin Zhou","doi":"10.26493/1855-3974.2122.1E2","DOIUrl":"https://doi.org/10.26493/1855-3974.2122.1E2","url":null,"abstract":"For a k -regular graph Γ and a graph Υ of order k , a generalized truncation of Γ by Υ is constructed by replacing each vertex of Γ with a copy of Υ . E. Eiben, R. Jajcay and P. S parl introduced a method for constructing vertex-transitive generalized truncations. For convenience, we call a graph obtained by using Eiben et al. ’s method a special generalized truncation . In their paper, Eiben et al. proposed a problem to classify special generalized truncations of a complete graph K n by a cycle of length n − 1 . In this paper, we completely solve this problem by demonstrating that with the exception of n = 6 , every special generalized truncation of a complete graph K n by a cycle of length n − 1 is a Cayley graph of AGL(1, n ) where n is a prime power. Moreover, the full automorphism groups of all these graphs and the isomorphisms among them are determined.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74001025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-15DOI: 10.26493/1855-3974.2125.7b0
D. Bartoli, G. Kiss, S. Marcugini, F. Pambianco
Lower and upper bounds on the size of resolving sets and semi-resolving sets for the point-line incidence graph of the finite projective space P G ( n , q ) are presented. It is proved that if n > 2 is fixed, then the metric dimension of the graph is asymptotically 2 q n − 1 .
{"title":"On resolving sets in the point-line incidence graph of PG(n, q)","authors":"D. Bartoli, G. Kiss, S. Marcugini, F. Pambianco","doi":"10.26493/1855-3974.2125.7b0","DOIUrl":"https://doi.org/10.26493/1855-3974.2125.7b0","url":null,"abstract":"Lower and upper bounds on the size of resolving sets and semi-resolving sets for the point-line incidence graph of the finite projective space P G ( n , q ) are presented. It is proved that if n > 2 is fixed, then the metric dimension of the graph is asymptotically 2 q n − 1 .","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86932561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-14DOI: 10.26493/1855-3974.2039.efc
S. Bonvicini, A. Vietri
Using a technique which is inspired by topology, we construct original examples of 3 - and 4 -edge critical graphs. The 3 -critical graphs cover all even orders starting from 26 ; the 4 -critical graphs cover all even orders starting from 20 and all the odd orders. In particular, the 3 -critical graphs are not isomorphic to the graphs provided by Goldberg for disproving the Critical Graph Conjecture. Using the same approach we also revisit the construction of some fundamental critical graphs, such as Goldberg’s infinite family of 3 -critical graphs, Chetwynd’s 4 -critical graph of order 16 and Fiol’s 4 -critical graph of order 18 .
{"title":"A Möbius-type gluing technique for obtaining edge-critical graphs","authors":"S. Bonvicini, A. Vietri","doi":"10.26493/1855-3974.2039.efc","DOIUrl":"https://doi.org/10.26493/1855-3974.2039.efc","url":null,"abstract":"Using a technique which is inspired by topology, we construct original examples of 3 - and 4 -edge critical graphs. The 3 -critical graphs cover all even orders starting from 26 ; the 4 -critical graphs cover all even orders starting from 20 and all the odd orders. In particular, the 3 -critical graphs are not isomorphic to the graphs provided by Goldberg for disproving the Critical Graph Conjecture. Using the same approach we also revisit the construction of some fundamental critical graphs, such as Goldberg’s infinite family of 3 -critical graphs, Chetwynd’s 4 -critical graph of order 16 and Fiol’s 4 -critical graph of order 18 .","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87273487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-13DOI: 10.26493/1855-3974.2033.974
Jasna Prezelj, Fabio Vlacci
We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs H ( d , 2) . We prove that monomials in variables z and w are always linearly independent as functions only in bidegrees ( p , 0), ( p , 1), (0, q ), (1, q ) and (2, 2).
{"title":"Divergence zero quaternionic vector fields and Hamming graphs","authors":"Jasna Prezelj, Fabio Vlacci","doi":"10.26493/1855-3974.2033.974","DOIUrl":"https://doi.org/10.26493/1855-3974.2033.974","url":null,"abstract":"We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs H ( d , 2) . We prove that monomials in variables z and w are always linearly independent as functions only in bidegrees ( p , 0), ( p , 1), (0, q ), (1, q ) and (2, 2).","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84874986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-12DOI: 10.26493/1855-3974.2137.7FA
D. Bartoli, Corrado Zanella, Ferdinando Zullo
We generalize the example of linear set presented by the last two authors in “Vertex properties of maximum scattered linear sets of PG (1, q n ) " (2019) to a more general family, proving that such linear sets are maximum scattered when q is odd and, apart from a special case, they are new. This solves an open problem posed in “Vertex properties of maximum scattered linear sets of PG (1, q n ) " (2019). As a consequence of Sheekey’s results in “A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters (6, 6, q ; 5) .
{"title":"A new family of maximum scattered linear sets in PG(1, q6)","authors":"D. Bartoli, Corrado Zanella, Ferdinando Zullo","doi":"10.26493/1855-3974.2137.7FA","DOIUrl":"https://doi.org/10.26493/1855-3974.2137.7FA","url":null,"abstract":"We generalize the example of linear set presented by the last two authors in “Vertex properties of maximum scattered linear sets of PG (1, q n ) \" (2019) to a more general family, proving that such linear sets are maximum scattered when q is odd and, apart from a special case, they are new. This solves an open problem posed in “Vertex properties of maximum scattered linear sets of PG (1, q n ) \" (2019). As a consequence of Sheekey’s results in “A new family of linear maximum rank distance codes\" (2016), this family yields to new MRD-codes with parameters (6, 6, q ; 5) .","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77290208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-10DOI: 10.26493/1855-3974.2318.FB9
Abel Cabrera Martínez, A. Estrada-Moreno, J. A. Rodríguez-Velázquez
In this paper, we show that the Italian domination number of every lexicographic product graph $Gcirc H$ can be expressed in terms of five different domination parameters of $G$. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let $N(v)$ denote the open neighbourhood of $vin V(G)$, and let $w=(w_0,w_1, dots,w_l)$ be a vector of nonnegative integers such that $ w_0ge 1$. We say that a function $f: V(G)longrightarrow {0,1,dots ,l}$ is a $w$-dominating function if $f(N(v))=sum_{uin N(v)}f(u)ge w_i$ for every vertex $v$ with $f(v)=i$. The weight of $f$ is defined to be $omega(f)=sum_{vin V(G)} f(v)$. The $w$-domination number of $G$, denoted by $gamma_{w}(G)$, is the minimum weight among all $w$-dominating functions on $G$. If we impose restrictions on the minimum degree of $G$ when needed, under this approach we can define, for instance, the domination number, the total domination number, the $k$-domination number, the $k$-tuple domination number, the $k$-tuple total domination number, the Italian domination number, the total Italian domination number, and the ${k}$-domination number. Specifically, we show that $gamma_{I}(Gcirc H)=gamma_{w}(G)$, where $win {2}times{0,1,2}^{l}$ and $lin {2,3}$. The decision on whether the equality holds for specific values of $w_0,dots,w_l$ will depend on the value of the domination number of $H$. This paper also provides preliminary results on $gamma_{w}(G)$ and raises the challenge of conducting a detailed study of the topic.
{"title":"From Italian domination in lexicographic product graphs to w-domination in graphs","authors":"Abel Cabrera Martínez, A. Estrada-Moreno, J. A. Rodríguez-Velázquez","doi":"10.26493/1855-3974.2318.FB9","DOIUrl":"https://doi.org/10.26493/1855-3974.2318.FB9","url":null,"abstract":"In this paper, we show that the Italian domination number of every lexicographic product graph $Gcirc H$ can be expressed in terms of five different domination parameters of $G$. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let $N(v)$ denote the open neighbourhood of $vin V(G)$, and let $w=(w_0,w_1, dots,w_l)$ be a vector of nonnegative integers such that $ w_0ge 1$. We say that a function $f: V(G)longrightarrow {0,1,dots ,l}$ is a $w$-dominating function if $f(N(v))=sum_{uin N(v)}f(u)ge w_i$ for every vertex $v$ with $f(v)=i$. The weight of $f$ is defined to be $omega(f)=sum_{vin V(G)} f(v)$. The $w$-domination number of $G$, denoted by $gamma_{w}(G)$, is the minimum weight among all $w$-dominating functions on $G$. If we impose restrictions on the minimum degree of $G$ when needed, under this approach we can define, for instance, the domination number, the total domination number, the $k$-domination number, the $k$-tuple domination number, the $k$-tuple total domination number, the Italian domination number, the total Italian domination number, and the ${k}$-domination number. Specifically, we show that $gamma_{I}(Gcirc H)=gamma_{w}(G)$, where $win {2}times{0,1,2}^{l}$ and $lin {2,3}$. The decision on whether the equality holds for specific values of $w_0,dots,w_l$ will depend on the value of the domination number of $H$. This paper also provides preliminary results on $gamma_{w}(G)$ and raises the challenge of conducting a detailed study of the topic.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76957112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-02DOI: 10.26493/1855-3974.2573.90d
Teeradej Kittipassorn, Panon Sinsap
In this paper, we prove that given a 2-edge-coloured complete graph $K_{4n}$ that has the same number of edges of each colour, we can always find a perfect matching with an equal number of edges of each colour. This solves a problem posed by Caro, Hansberg, Lauri, and Zarb. The problem is also independently solved by Ehard, Mohr, and Rautenbach.
{"title":"On the existence of zero-sum perfect matchings of complete graphs","authors":"Teeradej Kittipassorn, Panon Sinsap","doi":"10.26493/1855-3974.2573.90d","DOIUrl":"https://doi.org/10.26493/1855-3974.2573.90d","url":null,"abstract":"In this paper, we prove that given a 2-edge-coloured complete graph $K_{4n}$ that has the same number of edges of each colour, we can always find a perfect matching with an equal number of edges of each colour. This solves a problem posed by Caro, Hansberg, Lauri, and Zarb. The problem is also independently solved by Ehard, Mohr, and Rautenbach.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91102178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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