Pub Date : 2021-04-14DOI: 10.26493/1855-3974.2388.928
A. Muratovic-Ribic
We present an application of generalized strong complete mappings to construction of a family of mutually orthogonal Latin squares. We also determine a cycle structure of such mapping which form a complete family of MOLS. Many constructions of generalized strong complete mappings over an extension of finite field are provided.
{"title":"On generalized strong complete mappings and mutually orthogonal Latin squares","authors":"A. Muratovic-Ribic","doi":"10.26493/1855-3974.2388.928","DOIUrl":"https://doi.org/10.26493/1855-3974.2388.928","url":null,"abstract":"We present an application of generalized strong complete mappings to construction of a family of mutually orthogonal Latin squares. We also determine a cycle structure of such mapping which form a complete family of MOLS. Many constructions of generalized strong complete mappings over an extension of finite field are provided.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89191883","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 : 2021-04-13DOI: 10.26493/1855-3974.2467.497
A. Brodnik, Marko Grgurovic, Rok Požar
The paper describes two relatively simple modifications of the well-known FloydWarshall algorithm for computing all-pairs shortest paths. A fundamental difference of both modifications in comparison to the Floyd-Warshall algorithm is that the relaxation is done in a smart way. We show that the expected-case time complexity of both algorithms is O(n log n) for the class of complete directed graphs on n vertices with arc weights selected independently at random from the uniform distribution on [0, 1]. Theoretically best known algorithms for this class of graphs are all based on Dijkstra’s algorithm and obtain a better expected-case bound. However, by conducting an empirical evaluation we prove that our algorithms are at least competitive in practice with best know algorithms and, moreover, outperform most of them. The reason for the practical efficiency of the presented algorithms is the absence of use of priority queue. ∗A preliminary version of this work has been published in Shortest Path Solvers: From Software to Wetware, volume 32 of Emergence, Complexity and Computation (2018). The authors would like to thank the reviewer for excellent comments that substantially improved the quality of the paper. †This work is sponsored in part by the Slovenian Research Agency (research program P2-0359 and research projects J1-2481, J2-2504, and N2-0171). ‡Corresponding author. This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0062, J1-9110, J1-9187, J1-1694, N1-0159, J1-2451). cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ Ac ce pt ed m an us cr ip t 2 Ars Math. Contemp.
本文对众所周知的计算全对最短路径的FloydWarshall算法进行了两个相对简单的改进。与Floyd-Warshall算法相比，这两种修改的根本区别在于，松弛是以一种聪明的方式完成的。我们证明了这两种算法的期望情况时间复杂度都是O(n log n)，对于有n个顶点的完全有向图，这些顶点的弧权值是从均匀分布[0,1]中随机独立选择的。理论上，这类图最著名的算法都是基于Dijkstra算法，并获得更好的期望情况界。然而，通过进行经验评估，我们证明了我们的算法在实践中至少与最知名的算法具有竞争力，而且表现优于大多数算法。所提出的算法的实际效率的原因是没有使用优先级队列。*这项工作的初步版本已发表在最短路径求解器:从软件到湿软件，涌现，复杂性和计算(2018)第32卷。作者要感谢审稿人的优秀意见，这大大提高了论文的质量。†这项工作部分由斯洛文尼亚研究机构赞助(研究计划P2-0359和研究项目J1-2481, J2-2504和N2-0171)。‡通讯作者。这项工作得到了斯洛文尼亚研究机构的部分支持(研究项目P1-0285和研究项目N1-0062、J1-9110、J1-9187、J1-1694、N1-0159、J1-2451)。本作品在https://creativecommons.org/licenses/by/4.0/下获得许可，并由2 Ars Math授权。一栏。
{"title":"Modifications of the Floyd-Warshall algorithm with nearly quadratic expected-time","authors":"A. Brodnik, Marko Grgurovic, Rok Požar","doi":"10.26493/1855-3974.2467.497","DOIUrl":"https://doi.org/10.26493/1855-3974.2467.497","url":null,"abstract":"The paper describes two relatively simple modifications of the well-known FloydWarshall algorithm for computing all-pairs shortest paths. A fundamental difference of both modifications in comparison to the Floyd-Warshall algorithm is that the relaxation is done in a smart way. We show that the expected-case time complexity of both algorithms is O(n log n) for the class of complete directed graphs on n vertices with arc weights selected independently at random from the uniform distribution on [0, 1]. Theoretically best known algorithms for this class of graphs are all based on Dijkstra’s algorithm and obtain a better expected-case bound. However, by conducting an empirical evaluation we prove that our algorithms are at least competitive in practice with best know algorithms and, moreover, outperform most of them. The reason for the practical efficiency of the presented algorithms is the absence of use of priority queue. ∗A preliminary version of this work has been published in Shortest Path Solvers: From Software to Wetware, volume 32 of Emergence, Complexity and Computation (2018). The authors would like to thank the reviewer for excellent comments that substantially improved the quality of the paper. †This work is sponsored in part by the Slovenian Research Agency (research program P2-0359 and research projects J1-2481, J2-2504, and N2-0171). ‡Corresponding author. This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects N1-0062, J1-9110, J1-9187, J1-1694, N1-0159, J1-2451). cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ Ac ce pt ed m an us cr ip t 2 Ars Math. Contemp.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"26 1","pages":"1"},"PeriodicalIF":0.0,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73707447","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 : 2021-04-11DOI: 10.26493/1855-3974.2478.D1B
S. Elizalde, Rigoberto Flórez, J. L. Ramírez
{"title":"Enumerating symmetric peaks in non-decreasing Dyck paths","authors":"S. Elizalde, Rigoberto Flórez, J. L. Ramírez","doi":"10.26493/1855-3974.2478.D1B","DOIUrl":"https://doi.org/10.26493/1855-3974.2478.D1B","url":null,"abstract":"","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87097089","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 : 2021-04-07DOI: 10.26493/1855-3974.2600.dcc
Abel Cabrera Martínez
In a graph G, a vertex dominates itself and its neighbours. A set D ⊆ V (G) is said to be a k-tuple dominating set of G if D dominates every vertex of G at least k times. The minimum cardinality among all k-tuple dominating sets is the k-tuple domination number of G. In this note, we provide new bounds on this parameter. Some of these bounds generalize other ones that have been given for the case k = 2.
{"title":"A note on the k-tuple domination number of graphs","authors":"Abel Cabrera Martínez","doi":"10.26493/1855-3974.2600.dcc","DOIUrl":"https://doi.org/10.26493/1855-3974.2600.dcc","url":null,"abstract":"In a graph G, a vertex dominates itself and its neighbours. A set D ⊆ V (G) is said to be a k-tuple dominating set of G if D dominates every vertex of G at least k times. The minimum cardinality among all k-tuple dominating sets is the k-tuple domination number of G. In this note, we provide new bounds on this parameter. Some of these bounds generalize other ones that have been given for the case k = 2.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"8 1","pages":"4"},"PeriodicalIF":0.0,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74086572","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 : 2021-04-06DOI: 10.26493/1855-3974.2338.5DF
M. Noce
We construct a new example of an infinite family of groups acting on a d -adic tree, with d ≥ 2 that is non-contracting and weakly regular branch over the derived subgroup.
我们构造了作用于d矢树上的无限群族的一个新例子，其中d≥2是非收缩的弱正则分支。
{"title":"A family of fractal non-contracting weakly branch groups","authors":"M. Noce","doi":"10.26493/1855-3974.2338.5DF","DOIUrl":"https://doi.org/10.26493/1855-3974.2338.5DF","url":null,"abstract":"We construct a new example of an infinite family of groups acting on a d -adic tree, with d ≥ 2 that is non-contracting and weakly regular branch over the derived subgroup.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"44 1","pages":"29-36"},"PeriodicalIF":0.0,"publicationDate":"2021-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87550330","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 : 2021-03-12DOI: 10.26493/1855-3974.2359.A7B
Nobin Thomas, Lisa Mathew, S. Sriram, K. Subramanian
A new class of graphs G(w), called Parikh word representable graphs (PWRG), corresponding to words $w$ that are finite sequence of symbols, was considered in the recent past. Several properties of these graphs have been established. In this paper, we consider these graphs corresponding to binary core words of the form $aub$ over a binary alphabet {a,b}. We derive formulas for computing the Wiener index of the PWRG of a binary core word. Sharp bounds are established on the value of this index in terms of different parameters related to binary words over {a,b} and the corresponding PWRGs. Certain other Wiener-type indices that are variants of Wiener index are also considered. Formulas for computing these indices in the case of PWRG of a binary core word are obtained.
{"title":"Wiener-type indices of Parikh word representable graphs","authors":"Nobin Thomas, Lisa Mathew, S. Sriram, K. Subramanian","doi":"10.26493/1855-3974.2359.A7B","DOIUrl":"https://doi.org/10.26493/1855-3974.2359.A7B","url":null,"abstract":"A new class of graphs G(w), called Parikh word representable graphs (PWRG), corresponding to words $w$ that are finite sequence of symbols, was considered in the recent past. Several properties of these graphs have been established. In this paper, we consider these graphs corresponding to binary core words of the form $aub$ over a binary alphabet {a,b}. We derive formulas for computing the Wiener index of the PWRG of a binary core word. Sharp bounds are established on the value of this index in terms of different parameters related to binary words over {a,b} and the corresponding PWRGs. Certain other Wiener-type indices that are variants of Wiener index are also considered. Formulas for computing these indices in the case of PWRG of a binary core word are obtained.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"51 1","pages":"243-260"},"PeriodicalIF":0.0,"publicationDate":"2021-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79541992","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 : 2021-03-11DOI: 10.26493/1855-3974.2201.B65
K. Petelczyc, K. Prażmowski, M. Żynel
The concept of spine geometry over a polar Grassmann space was introduced in [9]. The geometry in question belongs also to a wide family of partial affine line spaces. It is known that such a geometry -- e.g. the ``ordinary'' spine geometry, as considered in [13, 14] can be developed in terms of points, so called affine lines, and their parallelism (in this case this parallelism is not intrinsically definable: it is not `Veblenian', cf. [11]). This paper aims to prove an analogous result for polar spine spaces. As a by-product we obtain several other results on primitive notions for the geometry of polar spine spaces.
{"title":"Geometry of the parallelism in polar spine spaces and their line reducts","authors":"K. Petelczyc, K. Prażmowski, M. Żynel","doi":"10.26493/1855-3974.2201.B65","DOIUrl":"https://doi.org/10.26493/1855-3974.2201.B65","url":null,"abstract":"The concept of spine geometry over a polar Grassmann space was introduced in [9]. The geometry in question belongs also to a wide family of partial affine line spaces. It is known that such a geometry -- e.g. the ``ordinary'' spine geometry, as considered in [13, 14] can be developed in terms of points, so called affine lines, and their parallelism (in this case this parallelism is not intrinsically definable: it is not `Veblenian', cf. [11]). This paper aims to prove an analogous result for polar spine spaces. As a by-product we obtain several other results on primitive notions for the geometry of polar spine spaces.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"39 1","pages":"151-170"},"PeriodicalIF":0.0,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77397341","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 : 2021-03-11DOI: 10.26493/1855-3974.2284.AEB
Abel Cabrera Martínez, J. A. Rodríguez-Velázquez
Let G be a graph with no isolated vertex and f : V ( G ) → {0, 1, 2} a function. Let V i = { x ∈ V ( G ) : f ( x ) = i } for every i ∈ {0, 1, 2} . We say that f is a total Roman dominating function on G if every vertex in V 0 is adjacent to at least one vertex in V 2 and the subgraph induced by V 1 ∪ V 2 has no isolated vertex. The weight of f is ω ( f ) = ∑ v ∈ V ( G ) f ( v ) . The minimum weight among all total Roman dominating functions on G is the total Roman domination number of G , denoted by γ t R ( G ) . It is known that the general problem of computing γ t R ( G ) is NP-hard. In this paper, we show that if G is a graph with no isolated vertex and H is a nontrivial graph, then the total Roman domination number of the lexicographic product graph G ∘ H is given by γ t R ( G ∘ H ) = 2 γ t ( G ) if γ ( H ) ≥ 2, and γ t R ( G ∘ H ) = ξ ( G ) if γ ( H ) = 1 , where γ ( H ) is the domination number of H , γ t ( G ) is the total domination number of G and ξ ( G ) is a domination parameter defined on G .
设G为无孤立顶点的图，f: V (G)→{0,1,2}为函数。让V i = {x∈V (G): f (x) =我}每我∈{0 1 2}。我们说f是G上的全罗马支配函数，如果v0中的每个顶点都与v2中的至少一个顶点相邻，并且由v1∪v2引出的子图没有孤立的顶点。f的权值为ω (f) =∑v∈v (G) f (v)。G上所有总罗马支配函数的最小权值是G的总罗马支配数，记为γ t R (G)。众所周知，计算γ t R (G)的一般问题是np困难的。在本文中,我们表明,如果G图没有孤立的顶点和H是一个重要的图,然后整个罗马统治的词典给出的产品图G∘H t R (G∘H) = 2γγt (G)如果γ(H)≥2,和γt R (G∘H) =ξ(G)如果γ(H) = 1,在γ(H)是统治的H,γt (G)是完全统治的G和ξ(G)是一个控制参数定义在G。
{"title":"Closed formulas for the total Roman domination number of lexicographic product graphs","authors":"Abel Cabrera Martínez, J. A. Rodríguez-Velázquez","doi":"10.26493/1855-3974.2284.AEB","DOIUrl":"https://doi.org/10.26493/1855-3974.2284.AEB","url":null,"abstract":"Let G be a graph with no isolated vertex and f : V ( G ) → {0, 1, 2} a function. Let V i = { x ∈ V ( G ) : f ( x ) = i } for every i ∈ {0, 1, 2} . We say that f is a total Roman dominating function on G if every vertex in V 0 is adjacent to at least one vertex in V 2 and the subgraph induced by V 1 ∪ V 2 has no isolated vertex. The weight of f is ω ( f ) = ∑ v ∈ V ( G ) f ( v ) . The minimum weight among all total Roman dominating functions on G is the total Roman domination number of G , denoted by γ t R ( G ) . It is known that the general problem of computing γ t R ( G ) is NP-hard. In this paper, we show that if G is a graph with no isolated vertex and H is a nontrivial graph, then the total Roman domination number of the lexicographic product graph G ∘ H is given by γ t R ( G ∘ H ) = 2 γ t ( G ) if γ ( H ) ≥ 2, and γ t R ( G ∘ H ) = ξ ( G ) if γ ( H ) = 1 , where γ ( H ) is the domination number of H , γ t ( G ) is the total domination number of G and ξ ( G ) is a domination parameter defined on G .","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"52 1","pages":"233-241"},"PeriodicalIF":0.0,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75954729","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 : 2021-03-04DOI: 10.26493/1855-3974.2173.71A
M. Pilsniak
The distinguishing index of a graph G, denoted by D′(G), is the least number of colours in an edge colouring of G not preserved by any nontrivial automorphism. This invariant is defined for any graph without K2 as a connected component and without two isolated vertices, and such a graph is called admissible. We prove the Nordhaus-Gaddum type relation: 2 ≤ D′(G) +D′(G) ≤ ∆(G) + 2 for every admissible connected graph G of order |G| ≥ 7 such that G is also admissible.
{"title":"Nordhaus-Gaddum type inequalities for the distinguishing index","authors":"M. Pilsniak","doi":"10.26493/1855-3974.2173.71A","DOIUrl":"https://doi.org/10.26493/1855-3974.2173.71A","url":null,"abstract":"The distinguishing index of a graph G, denoted by D′(G), is the least number of colours in an edge colouring of G not preserved by any nontrivial automorphism. This invariant is defined for any graph without K2 as a connected component and without two isolated vertices, and such a graph is called admissible. We prove the Nordhaus-Gaddum type relation: 2 ≤ D′(G) +D′(G) ≤ ∆(G) + 2 for every admissible connected graph G of order |G| ≥ 7 such that G is also admissible.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":"142 1","pages":"223-231"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87965385","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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