Pub Date : 2020-11-02DOI: 10.26493/1855-3974.2034.dad
Wasim Abbas, M. Hirasaka
In this article, we focus on association schemes with some properties derived from the orbitals of a transitive permutation group G with a one-point stabilizer H satisfying H < N G ( H ) < N G ( N G ( H )) ⊴ G and | N G ( N G ( H ))| = p 3 where p is a prime. By a corollary of our main result we obtain some inequality which corresponds to the fact | G : N G ( N G ( H ))| ≤ p + 1 .
在这个文章,我们专注在协会计划能与一些财产之orbitals derived from a transitive permutation集团G和H a one-point稳定器令人满意G H < N (H) < N G G (N (H)⊴G和G | N (N G p (H) | = p 3哪里是一个撇号。By a corollary of our玩》的论点我们得到一些不平等哪种corresponds的| G G: N (N G (H) |≤p + 1。
{"title":"Association schemes with a certain type of p-subschemes","authors":"Wasim Abbas, M. Hirasaka","doi":"10.26493/1855-3974.2034.dad","DOIUrl":"https://doi.org/10.26493/1855-3974.2034.dad","url":null,"abstract":"In this article, we focus on association schemes with some properties derived from the orbitals of a transitive permutation group G with a one-point stabilizer H satisfying H < N G ( H ) < N G ( N G ( H )) ⊴ G and | N G ( N G ( H ))| = p 3 where p is a prime. By a corollary of our main result we obtain some inequality which corresponds to the fact | G : N G ( N G ( H ))| ≤ p + 1 .","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":"86078798","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: 10.26493/1855-3974.2106.423
Tamás Héger, Lisa Hernandez Lucas
A dominating set in a graph is a set of vertices such that each vertex not in the set has a neighbor in the set. The domination number is the smallest size of a dominating set. We consider this problem in the incidence graph of a generalized quadrangle. We show that the domination number of a generalized quadrangle with parameters s and t is at most 2 s t + 1 , and we prove that this bound is sharp if s = t or if s = q − 1 and t = q + 1 . Moreover, we give a complete classification of smallest dominating sets in generalized quadrangles where s = t , and give some general results for small dominating sets in the general case.
{"title":"Dominating sets in finite generalized quadrangles","authors":"Tamás Héger, Lisa Hernandez Lucas","doi":"10.26493/1855-3974.2106.423","DOIUrl":"https://doi.org/10.26493/1855-3974.2106.423","url":null,"abstract":"A dominating set in a graph is a set of vertices such that each vertex not in the set has a neighbor in the set. The domination number is the smallest size of a dominating set. We consider this problem in the incidence graph of a generalized quadrangle. We show that the domination number of a generalized quadrangle with parameters s and t is at most 2 s t + 1 , and we prove that this bound is sharp if s = t or if s = q − 1 and t = q + 1 . Moreover, we give a complete classification of smallest dominating sets in generalized quadrangles where s = t , and give some general results for small dominating sets in the general case.","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":"85571605","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 : 2020-10-28DOI: 10.26493/1855-3974.2028.cb4
Zülfükar Saygı
Lucas cubes are the special subgraphs of Fibonacci cubes. For small dimensions, their domination numbers are obtained by direct search or integer linear programming. For larger dimensions some bounds on these numbers are given. In this work, we present the exact values of total domination number of small dimensional Lucas cubes and present optimization problems obtained from the degree information of Lucas cubes, whose solutions give better lower bounds on the domination numbers and total domination numbers of Lucas cubes.
{"title":"Results on the domination number and the total domination number of Lucas cubes","authors":"Zülfükar Saygı","doi":"10.26493/1855-3974.2028.cb4","DOIUrl":"https://doi.org/10.26493/1855-3974.2028.cb4","url":null,"abstract":"Lucas cubes are the special subgraphs of Fibonacci cubes. For small dimensions, their domination numbers are obtained by direct search or integer linear programming. For larger dimensions some bounds on these numbers are given. In this work, we present the exact values of total domination number of small dimensional Lucas cubes and present optimization problems obtained from the degree information of Lucas cubes, whose solutions give better lower bounds on the domination numbers and total domination numbers of Lucas cubes.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74397674","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 : 2020-10-26DOI: 10.26493/1855-3974.2163.5df
Shao-Fei Du, Klavdija Kutnar, D. Marušič
A step forward is made in a long standing Lovasz problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes arising from the group action of the projective special linear group PSL (2, q 2 ) on cosets of its subgroup isomorphic to the projective general linear group PGL (2, q ) contains a Hamilton cycle.
{"title":"Hamilton cycles in primitive vertex-transitive graphs of order a product of two primes - the case PSL(2, q2) acting on cosets of PGL(2, q)","authors":"Shao-Fei Du, Klavdija Kutnar, D. Marušič","doi":"10.26493/1855-3974.2163.5df","DOIUrl":"https://doi.org/10.26493/1855-3974.2163.5df","url":null,"abstract":"A step forward is made in a long standing Lovasz problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes arising from the group action of the projective special linear group PSL (2, q 2 ) on cosets of its subgroup isomorphic to the projective general linear group PGL (2, q ) contains a Hamilton cycle.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77636170","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 : 2020-10-24DOI: 10.26493/1855-3974.2239.7F1
Tomoki Nakamigawa
A chord diagram E is a set of chords of a circle such that no pair of chords has a common endvertex. Let v 1 , v 2 , …, v 2 n be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram { v 1 v n + 1 , v 2 v n + 2 , …, v n v 2 n } is called an n -crossing and a chord diagram { v 1 v 2 , v 3 v 4 , …, v 2 n − 1 v 2 n } is called an n -necklace. For a chord diagram E having a 2 -crossing S = { x 1 x 3 , x 2 x 4 } , the expansion of E with respect to S is to replace E with E 1 = ( E S ) ∪ { x 2 x 3 , x 4 x 1 } or E 2 = ( E S ) ∪ { x 1 x 2 , x 3 x 4 } . Beginning from a given chord diagram E as the root, by iterating chord expansions in both ways, we have a binary tree whose all leaves are nonintersecting chord diagrams. Let NCD ( E ) be the multiset of the leaves. In this paper, the multiplicity of an n -necklace in NCD ( E ) is studied. Among other results, it is shown that the multiplicity of an n -necklace generated from an n -crossing equals the Genocchi number when n is odd and the median Genocchi number when n is even.
弦图E是一个圆的一组弦，没有一对弦有一个共同的端点。设v1, v2，…，v2 n是沿圆周顺时针排列的一系列顶点。弦图{v1 v n + 1, v2 v n + 2，…，v n v 2n}称为n交叉，弦图{v1 v2, v3 v4，…，v2 n−1 v2 n}称为n项链。对于一个2叉弦图E = {x 1 x 3, x 2 x 4}， E关于S的展开式是将E替换为e1 = (E S)∪{x 2 x 3, x 4 x 1}或e2 = (E S)∪{x 1 x 2, x 3 x 4}。从给定的和弦图E作为根开始，通过两种方式的迭代和弦展开，我们得到了一棵二叉树，它的所有叶都是不相交的和弦图。设NCD (E)为叶的多集。本文研究了NCD (E)中n -项链的多重性。在其他结果中，证明了由n交叉生成的n项链的多重性等于n为奇数时的genochi数和n为偶数时的中位数genochi数。
{"title":"The expansion of a chord diagram and the Genocchi numbers","authors":"Tomoki Nakamigawa","doi":"10.26493/1855-3974.2239.7F1","DOIUrl":"https://doi.org/10.26493/1855-3974.2239.7F1","url":null,"abstract":"A chord diagram E is a set of chords of a circle such that no pair of chords has a common endvertex. Let v 1 , v 2 , …, v 2 n be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram { v 1 v n + 1 , v 2 v n + 2 , …, v n v 2 n } is called an n -crossing and a chord diagram { v 1 v 2 , v 3 v 4 , …, v 2 n − 1 v 2 n } is called an n -necklace. For a chord diagram E having a 2 -crossing S = { x 1 x 3 , x 2 x 4 } , the expansion of E with respect to S is to replace E with E 1 = ( E S ) ∪ { x 2 x 3 , x 4 x 1 } or E 2 = ( E S ) ∪ { x 1 x 2 , x 3 x 4 } . Beginning from a given chord diagram E as the root, by iterating chord expansions in both ways, we have a binary tree whose all leaves are nonintersecting chord diagrams. Let NCD ( E ) be the multiset of the leaves. In this paper, the multiplicity of an n -necklace in NCD ( E ) is studied. Among other results, it is shown that the multiplicity of an n -necklace generated from an n -crossing equals the Genocchi number when n is odd and the median Genocchi number when n is even.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73254152","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 : 2020-10-23DOI: 10.26493/1855-3974.1900.cc1
Xue Yu, B. Lou, Wenwen Fan
In 2018, Fan and Li classified the complete bipartite graph K m , n that has a unique orientably edge-transitive embedding. In this paper, we extend this to give a complete classification of K m , n which have exactly two orientably edge-transitive embeddings.
2018年，Fan和Li分类了具有唯一可定向边传递嵌入的完全二部图K m, n。在本文中，我们推广了这一理论，给出了K m n的完全分类，其中K m n恰好有两个可定向的边传递嵌入。
{"title":"The complete bipartite graphs which have exactly two orientably edge-transitive embeddings","authors":"Xue Yu, B. Lou, Wenwen Fan","doi":"10.26493/1855-3974.1900.cc1","DOIUrl":"https://doi.org/10.26493/1855-3974.1900.cc1","url":null,"abstract":"In 2018, Fan and Li classified the complete bipartite graph K m , n that has a unique orientably edge-transitive embedding. In this paper, we extend this to give a complete classification of K m , n which have exactly two orientably edge-transitive embeddings.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79871160","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 : 2020-10-21DOI: 10.26493/1855-3974.2019.30b
Mark Budden
In this paper, we introduce two different generalizations of Schur numbers that involve rainbow colorings. Motivated by well-known generalizations of Ramsey numbers, we first define the rainbow Schur number R S ( n ) to be the minimum number of colors needed such that every coloring of {1, 2, …, n } , in which all available colors are used, contains a rainbow solution to a + b = c . It is shown that $$RS(n)=floor{log _2(n)}+2, quad mbox{for all } nge 3.$$ Second, we consider the Gallai-Schur number G S ( n ) , defined to be the least natural number such that every n -coloring of {1, 2, …, G S ( n )} that lacks rainbow solutions to the equation a + b = c necessarily contains a monochromatic solution to this equation. By connecting this number with the n -color Gallai-Ramsey number for triangles, it is shown that for all n ≥ 3 , $$GS(n)=left{ begin{array}{ll} 5^k & mbox{if} n=2k 2cdot 5^k & mbox{if} n=2k+1.end{array} right.$$
本文介绍了涉及彩虹着色的舒尔数的两种不同的推广。根据Ramsey数的著名推广，我们首先定义彩虹舒尔数rs (n)为所需的最小颜色数，使得每一种颜色 {1, 2，…，n } ，其中包含a + b = c的彩虹解决方案。结果表明 $$RS(n)=floor{log _2(n)}+2, quad mbox{for all } nge 3.$$ 其次，我们考虑Gallai-Schur数G S (n)，它被定义为最小的自然数，使得每一个n -着色 {1, 2，…，G S (n)} 缺少方程a + b = c的彩虹解必然包含这个方程的单色解。通过将该数与三角形的n色Gallai-Ramsey数联系起来，表明对于所有n≥3， $$GS(n)=left{ begin{array}{ll} 5^k & mbox{if} n=2k 2cdot 5^k & mbox{if} n=2k+1.end{array} right.$$
{"title":"Schur numbers involving rainbow colorings","authors":"Mark Budden","doi":"10.26493/1855-3974.2019.30b","DOIUrl":"https://doi.org/10.26493/1855-3974.2019.30b","url":null,"abstract":"In this paper, we introduce two different generalizations of Schur numbers that involve rainbow colorings. Motivated by well-known generalizations of Ramsey numbers, we first define the rainbow Schur number R S ( n ) to be the minimum number of colors needed such that every coloring of {1, 2, …, n } , in which all available colors are used, contains a rainbow solution to a + b = c . It is shown that $$RS(n)=floor{log _2(n)}+2, quad mbox{for all } nge 3.$$ Second, we consider the Gallai-Schur number G S ( n ) , defined to be the least natural number such that every n -coloring of {1, 2, …, G S ( n )} that lacks rainbow solutions to the equation a + b = c necessarily contains a monochromatic solution to this equation. By connecting this number with the n -color Gallai-Ramsey number for triangles, it is shown that for all n ≥ 3 , $$GS(n)=left{ begin{array}{ll} 5^k & mbox{if} n=2k 2cdot 5^k & mbox{if} n=2k+1.end{array} right.$$","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76508778","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 : 2020-10-20DOI: 10.26493/1855-3974.2022.44A
A. Klobučar, A. Klobucar
Double Roman domination is a stronger version of Roman domination that doubles the protection. The areas now have 0 , 1 , 2 or 3 legions. Every attacked area needs 2 legions for its defence, either their own, or borrowed from 1 or 2 neighbouring areas, which still have to keep at least 1 legion to themselves. The minimal number of legions in all areas together is equal to the double Roman domination number. In this paper we determine an upper bound and a lower bound for double Roman domination numbers on cardinal product of any two graphs. Also we determine the exact values of double Roman domination numbers on P 2 × G (for many types of graph G ). Also, the double Roman domination number is found for P 2 × P n , P 3 × P n , P 4 × P n , while upper and lower bounds are given for P 5 × P n and P 6 × P n . Finally, we will give a case study to determine the efficiency of double protection. We will compare double Roman domination versus Roman domination by running a simulation of a battle.
{"title":"Properties of double Roman domination on cardinal products of graphs","authors":"A. Klobučar, A. Klobucar","doi":"10.26493/1855-3974.2022.44A","DOIUrl":"https://doi.org/10.26493/1855-3974.2022.44A","url":null,"abstract":"Double Roman domination is a stronger version of Roman domination that doubles the protection. The areas now have 0 , 1 , 2 or 3 legions. Every attacked area needs 2 legions for its defence, either their own, or borrowed from 1 or 2 neighbouring areas, which still have to keep at least 1 legion to themselves. The minimal number of legions in all areas together is equal to the double Roman domination number. In this paper we determine an upper bound and a lower bound for double Roman domination numbers on cardinal product of any two graphs. Also we determine the exact values of double Roman domination numbers on P 2 × G (for many types of graph G ). Also, the double Roman domination number is found for P 2 × P n , P 3 × P n , P 4 × P n , while upper and lower bounds are given for P 5 × P n and P 6 × P n . Finally, we will give a case study to determine the efficiency of double protection. We will compare double Roman domination versus Roman domination by running a simulation of a battle.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86279637","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 : 2020-10-19DOI: 10.26493/1855-3974.1793.C4D
Gábor Nyul, G. Rácz
The total number of partitions of a finite set into nonempty ordered subsets such that r distinguished elements belong to distinct ordered blocks can be described as sums of r -Lah numbers. In this paper we study this possible variant of Bell-like numbers, as well as the related r -Lah polynomials.
{"title":"Sums of r-Lah numbers and r-Lah polynomials","authors":"Gábor Nyul, G. Rácz","doi":"10.26493/1855-3974.1793.C4D","DOIUrl":"https://doi.org/10.26493/1855-3974.1793.C4D","url":null,"abstract":"The total number of partitions of a finite set into nonempty ordered subsets such that r distinguished elements belong to distinct ordered blocks can be described as sums of r -Lah numbers. In this paper we study this possible variant of Bell-like numbers, as well as the related r -Lah polynomials.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90404879","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 : 2020-10-19DOI: 10.26493/1855-3974.1931.9CF
Pieter Goetschalckx, K. Coolsaet, N. Cleemput
We introduce a new practical and more general definition of local symmetry-preserving operations on polyhedra. These can be applied to arbitrary embedded graphs and result in embedded graphs with the same or higher symmetry. With some additional properties we can restrict the connectivity, e.g. when we only want to consider polyhedra. Using some base structures and a list of 10 extensions, we can generate all possible local symmetry-preserving operations isomorph-free.
{"title":"Generation of local symmetry-preserving operations on polyhedra","authors":"Pieter Goetschalckx, K. Coolsaet, N. Cleemput","doi":"10.26493/1855-3974.1931.9CF","DOIUrl":"https://doi.org/10.26493/1855-3974.1931.9CF","url":null,"abstract":"We introduce a new practical and more general definition of local symmetry-preserving operations on polyhedra. These can be applied to arbitrary embedded graphs and result in embedded graphs with the same or higher symmetry. With some additional properties we can restrict the connectivity, e.g. when we only want to consider polyhedra. Using some base structures and a list of 10 extensions, we can generate all possible local symmetry-preserving operations isomorph-free.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78483137","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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