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":"40 1","pages":""},"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":"172 1","pages":""},"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":"46 1","pages":""},"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":"8 1","pages":""},"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":"17 1","pages":""},"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":"91 1","pages":""},"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":"9 1","pages":""},"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}
Pub Date : 2019-08-13DOI: 10.26493/1855-3974.2126.5b3
David Dolvzan, Gabriel Verret
We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.
在有限个零因子的反交换半环上，确定了矩阵的半环的零因子有向图的自同构群。
{"title":"The automorphism group of the zero-divisor digraph of matrices over an antiring","authors":"David Dolvzan, Gabriel Verret","doi":"10.26493/1855-3974.2126.5b3","DOIUrl":"https://doi.org/10.26493/1855-3974.2126.5b3","url":null,"abstract":"We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"14 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81090138","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 : 2019-05-16DOI: 10.26493/1855-3974.2677.b7f
Dániel Gerbner
We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.
{"title":"The covering lemma and q-analogues of extremal set theory problems","authors":"Dániel Gerbner","doi":"10.26493/1855-3974.2677.b7f","DOIUrl":"https://doi.org/10.26493/1855-3974.2677.b7f","url":null,"abstract":"We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"30 7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82746005","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 : 2018-06-25DOI: 10.26493/1855-3974.989.d15
Aureliano M. Robles Pérez, José Carlos Rosales
Let a = ( a 1 , …, a n ) and b = ( b 1 , …, b n ) be two n -tuples of positive integers, let X be a set of positive integers, and let g be a positive integer. In this work we show an algorithmic process in order to compute all the sets C of positive integers that fulfill the following conditions: The cardinality of C is equal to g ; If x , y ∈ ℕ {0} and x + y ∈ C , then C ∩ { x , y } ≠ ∅ ; If x ∈ C and ( x − b i ) / a i ∈ ℕ {0} for some i ∈ {1, …, n } , then ( x − b i ) / a i ∈ C ; X ∩ C = ∅ .
{"title":"A combinatorial problem and numerical semigroups","authors":"Aureliano M. Robles Pérez, José Carlos Rosales","doi":"10.26493/1855-3974.989.d15","DOIUrl":"https://doi.org/10.26493/1855-3974.989.d15","url":null,"abstract":"Let a = ( a 1 , …, a n ) and b = ( b 1 , …, b n ) be two n -tuples of positive integers, let X be a set of positive integers, and let g be a positive integer. In this work we show an algorithmic process in order to compute all the sets C of positive integers that fulfill the following conditions: The cardinality of C is equal to g ; If x , y ∈ ℕ {0} and x + y ∈ C , then C ∩ { x , y } ≠ ∅ ; If x ∈ C and ( x − b i ) / a i ∈ ℕ {0} for some i ∈ {1, …, n } , then ( x − b i ) / a i ∈ C ; X ∩ C = ∅ .","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"106 12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89625534","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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