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Maps and Δ-matroids revisited 地图和Δ-matroids重新访问
Art Discret. Appl. Math. Pub Date : 2020-09-22 DOI: 10.26493/2590-9770.1360.465
R. C. Avohou, B. Servatius, Herman Servatius
{"title":"Maps and Δ-matroids revisited","authors":"R. C. Avohou, B. Servatius, Herman Servatius","doi":"10.26493/2590-9770.1360.465","DOIUrl":"https://doi.org/10.26493/2590-9770.1360.465","url":null,"abstract":"Using Tutte’s combinatorial definition of a map we define a Δ-matroid purely combinatorially and show that it is identical to Bouchet’s topological definition.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125190884","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}
引用次数: 2
C_4-face-magic toroidal labelings on C_m × C_n C_m × C_n上的c_4面幻环面标记
Art Discret. Appl. Math. Pub Date : 2020-09-22 DOI: 10.26493/2590-9770.1368.F37
S. Curran, R. Low, S. C. Locke
{"title":"C_4-face-magic toroidal labelings on C_m × C_n","authors":"S. Curran, R. Low, S. C. Locke","doi":"10.26493/2590-9770.1368.F37","DOIUrl":"https://doi.org/10.26493/2590-9770.1368.F37","url":null,"abstract":"For a graph G = (V, E) naturally embedded in the torus, let ℱ(G) denote the set of faces of G. Then, G is called a Cn-face-magic toroidal graph if there exists a bijection f : V(G) → {1, 2, …, |V(G)|} such that for every F ∈ ℱ(G) with F ≅ Cn, the sum of all the vertex labels along Cn is a constant S. Let xv = f(v) for all v ∈ V(G). We call {xv : v ∈ V(G)} a Cn-face-magic toroidal labeling on G. We show that, for all m, n ≥ 2, Cm × Cn admits a C4-face-magic toroidal labeling if and only if either m = 2, or n = 2, or both m and n are even. We say that a C4-face-magic toroidal labeling {xi, j : (i, j) ∈ V(C2m × C2n)} on C2m × C2n is antipodal balanced if $x_{i,j} + x_{i+m,j+n} = tfrac{1}{2} S$, for all (i, j) ∈ V(C2m × C2n). We show that there exists an antipodal balanced C4-face-magic toroidal labeling on C2m × C2n if and only if the parity of m and n are the same. Furthermore, when both m and n are even, an antipodal balanced C4-face-magic toroidal labeling on C2m × C2n is both row-sum balanced and column-sum balanced. In addition, when m = n is even, an antipodal balanced C4-face-magic toroidal labeling on C2n × C2n is diagonal-sum balanced.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128189717","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}
引用次数: 0
A note on a candy sharing game 关于分享糖果游戏的笔记
Art Discret. Appl. Math. Pub Date : 2020-09-10 DOI: 10.26493/2590-9770.1304.2CF
Deepak Bal, Joseph DeGaetani
{"title":"A note on a candy sharing game","authors":"Deepak Bal, Joseph DeGaetani","doi":"10.26493/2590-9770.1304.2CF","DOIUrl":"https://doi.org/10.26493/2590-9770.1304.2CF","url":null,"abstract":"Suppose k students sit in a circle and are each distributed some initial amount of candy. Each student begins with an even amount of candy, but their individual amounts may vary. Upon the teacher’s signal, each student passes half of their candy to their left and keeps half. After this step, any student with an odd amount of candy receives an extra piece. The game ends if all the students are holding the same amount of candy. We prove, in a generalized setting, that for any initial distribution of n pieces of candy, the game terminates after O(log n) many iterations and each student ends with n/k + O(log n) many pieces. Moreover, there exist initial distributions for which the O(log n) term cannot be improved.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130485236","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}
引用次数: 0
Palindromic products 推销自己的产品
Art Discret. Appl. Math. Pub Date : 2020-09-08 DOI: 10.26493/2590-9770.1312.5b4
R. Hammack, Jamie Shive
{"title":"Palindromic products","authors":"R. Hammack, Jamie Shive","doi":"10.26493/2590-9770.1312.5b4","DOIUrl":"https://doi.org/10.26493/2590-9770.1312.5b4","url":null,"abstract":"","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121305488","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}
引用次数: 1
Special issue of ADAM devoted to the International Workshop on Symmetries of Graph and Networks 2018 ADAM专刊，致力于2018年图和网络对称性国际研讨会
Art Discret. Appl. Math. Pub Date : 2020-09-03 DOI: 10.26493/2590-9770.1384.0c9
T. Tucker, Yan-Quan Feng
{"title":"Special issue of ADAM devoted to the International Workshop on Symmetries of Graph and Networks 2018","authors":"T. Tucker, Yan-Quan Feng","doi":"10.26493/2590-9770.1384.0c9","DOIUrl":"https://doi.org/10.26493/2590-9770.1384.0c9","url":null,"abstract":"","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133265296","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}
引用次数: 0
On generalised Petersen graphs of girth 7 that have cop number 4 周长为7的广义Petersen图上的系数为4
Art Discret. Appl. Math. Pub Date : 2020-09-01 DOI: 10.26493/2590-9770.1382.2ad
Harmony Morris, Joy Morris
{"title":"On generalised Petersen graphs of girth 7 that have cop number 4","authors":"Harmony Morris, Joy Morris","doi":"10.26493/2590-9770.1382.2ad","DOIUrl":"https://doi.org/10.26493/2590-9770.1382.2ad","url":null,"abstract":"We show that if $n=7k/i$ with $i in {1,2,3}$ then the cop number of the generalised Petersen graph $GP(n,k)$ is $4$, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most $4$. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number $4$ but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number $4$ and girth $8$, while this paper explains those that have girth $7$.)","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123906939","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}
引用次数: 0
Bracing frameworks consisting of parallelograms 由平行四边形组成的支撑框架
Art Discret. Appl. Math. Pub Date : 2020-08-26 DOI: 10.26493/2590-9770.1379.7a4
Georg Grasegger, Jan Legerský
{"title":"Bracing frameworks consisting of parallelograms","authors":"Georg Grasegger, Jan Legerský","doi":"10.26493/2590-9770.1379.7a4","DOIUrl":"https://doi.org/10.26493/2590-9770.1379.7a4","url":null,"abstract":"A rectangle in the plane can be continuously deformed preserving its edge lengths, but adding a diagonal brace prevents such a deformation. Bolker and Crapo characterized combinatorially which choices of braces make a grid of squares infinitesimally rigid using a bracing graph: a bipartite graph whose vertices are the columns and rows of the grid, and a row and column are adjacent if and only if they meet at a braced square. Duarte and Francis generalized the notion of the bracing graph to rhombic carpets, proved that the connectivity of the bracing graph implies rigidity and stated the other implication without proof. Nagy Kem gives the equivalence in the infinitesimal setting. We consider continuous deformations of braced frameworks consisting of a graph from a more general class and its placement in the plane such that every 4-cycle forms a parallelogram. We show that rigidity of such a braced framework is equivalent to the non-existence of a special edge coloring, which is in turn equivalent to the corresponding bracing graph being connected.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116444154","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}
引用次数: 4
Sphere decompositions of hypercubes 超立方体的球分解
Art Discret. Appl. Math. Pub Date : 2020-08-23 DOI: 10.26493/2590-9770.1285.fd8
R. Hammack, P. C. Kainen
{"title":"Sphere decompositions of hypercubes","authors":"R. Hammack, P. C. Kainen","doi":"10.26493/2590-9770.1285.fd8","DOIUrl":"https://doi.org/10.26493/2590-9770.1285.fd8","url":null,"abstract":"For d ≡ 1 or 3 (mod 6), the 2-skeleton of the d-dimensional hypercube is decomposed into the union of pairwise face-disjoint isomorphic 2-complexes, each a topological sphere. If d = 5, then such a decomposition can be achieved, but with non-isomorphic spheres.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122594374","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}
引用次数: 3
Strongly regular graphs with parameters (37, 18, 8, 9) having nontrivial automorphisms 具有非平凡自同构参数(37,18,8,9)的强正则图
Art Discret. Appl. Math. Pub Date : 2020-08-21 DOI: 10.26493/2590-9770.1295.25b
D. Crnković, Marija Maksimović
{"title":"Strongly regular graphs with parameters (37, 18, 8, 9) having nontrivial automorphisms","authors":"D. Crnković, Marija Maksimović","doi":"10.26493/2590-9770.1295.25b","DOIUrl":"https://doi.org/10.26493/2590-9770.1295.25b","url":null,"abstract":"All strongly regular graphs having at most 36 vetrices have been enumerated. Hence, the first open case is enumeration of the SRGs with parameters (37,18,8,9). In this paper we show that there are exactly forty SRGs with parameters (37,18,8,9) having nontrivial automorphisms. Comparing the constructed graphs with previously known SRGs with these parameters we conclude that six of the SRGs with parameters (37,18,8,9) constructed in this paper are new, and that up to isomorphism there are at least 6766 strongly regular graphs with parameters (37,18,8,9).","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133829588","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}
引用次数: 2
Smart elements in combinatorial group testing problems with more defectives 组合组测试中智能元素缺陷较多的问题
Art Discret. Appl. Math. Pub Date : 2020-08-14 DOI: 10.26493/2590-9770.1363.b06
Máté Vizer, Dániel Gerbner
{"title":"Smart elements in combinatorial group testing problems with more defectives","authors":"Máté Vizer, Dániel Gerbner","doi":"10.26493/2590-9770.1363.b06","DOIUrl":"https://doi.org/10.26493/2590-9770.1363.b06","url":null,"abstract":"In combinatorial group testing problems Questioner needs to find a defective element x ∈ [ n ] by testing subsets of [ n ] . In [18] the authors introduced a new model, where each element knows the answer for those queries that contain it and each element should be able to identify the defective one. In this article we continue to investigate this kind of models with more defective elements. We also consider related models inspired by secret sharing models, where the elements should share information among them to find out the defectives. Finally the adaptive versions of the different models are also investigated.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"207 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133929337","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}
引用次数: 0
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