For a graph, a function is called an edge product cordial labeling of G, if the induced vertex labeling function is defined by the product of the labels of the incident edges as such that the number of edges with label 1 and the number of edges with label 0 differ by at most 1 and the number of vertices with label 1 and the number of vertices with label 0 differ by at most 1. In this paper, we show that the graphs obtained by duplication of a vertex, duplication of a vertex by an edge or duplication of an edge by a vertex in a crown graph are edge product cordial. Moreover, we show that the graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is edge product cordial. We also show that the graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial.
{"title":"Some Edge Product Cordial Graphs in the Context of Duplication of Some Graph Elements","authors":"U. M. Prajapati, Prakruti Shah","doi":"10.4236/OJDM.2016.64021","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64021","url":null,"abstract":"For a graph, a function is called an edge product cordial labeling of G, if the induced vertex labeling function is defined by the product of the labels of the incident edges as such that the number of edges with label 1 and the number of edges with label 0 differ by at most 1 and the number of vertices with label 1 and the number of vertices with label 0 differ by at most 1. In this paper, we show that the graphs obtained by duplication of a vertex, duplication of a vertex by an edge or duplication of an edge by a vertex in a crown graph are edge product cordial. Moreover, we show that the graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is edge product cordial. We also show that the graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"248-258"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626588","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}
One of the mainly interesting things of matroid theory is the representability of a matroid. Finding the set of all excluded minors for the representability is the solution of the representability. In 2000, Geelen, Gerards and Kapoor proved that is the complete set of GF(4)-representability. In this paper, we show that is an excluded minor for GF(4)-representability.
{"title":"Excluded Minority of P8 for GF(4)-Representability","authors":"S. Ahn, B. Han","doi":"10.4236/OJDM.2016.64024","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64024","url":null,"abstract":"One of the mainly interesting things of matroid theory is the representability of a matroid. Finding the set of all excluded minors for the representability is the solution of the representability. In 2000, Geelen, Gerards and Kapoor proved that is the complete set of GF(4)-representability. In this paper, we show that is an excluded minor for GF(4)-representability.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"279-296"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627279","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}
To enumerate isomers of the fluxional molecules, some theorems for maturity and the integer-valued characters of finite groups were introduced by S. Fujita and first author. The full non-rigid group of hexamethylethane is the semi-direct product of the direct products of six copies of the cyclic group Z3 by the dihedral group of order 12 (see, Asian J. Chem. (2010) 22 (3), 1966-1972). In this paper, we continue our study on finite groups (see Int. J. Theo. Physics, Group Theory, and Nonlinear Optics (2013), 17) and all the integer-valued characters of the above molecule are successfully derived.
为了列举流动分子的同分异构体，引入了有限群的成熟定理和整数值性质。六甲基乙烷的完整非刚性基团是12阶二面体基团Z3的六个副本的直接产物的半直接产物(见，Asian J. Chem。(2010) 22(3)， 1966-1972)。在本文中，我们继续对有限群的研究。j·西奥。《物理群论与非线性光学》(2013)，17)，成功推导出上述分子的所有整数值特征。
{"title":"The Z-Valued Characters for the Huge Symmetry of Hexamethylethane","authors":"A. Moghani, John P. Najarian","doi":"10.4236/OJDM.2016.64026","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64026","url":null,"abstract":"To enumerate isomers of the fluxional molecules, some theorems for maturity and the integer-valued characters of finite groups were introduced by S. Fujita and first author. The full non-rigid group of hexamethylethane is the semi-direct product of the direct products of six copies of the cyclic group Z3 by the dihedral group of order 12 (see, Asian J. Chem. (2010) 22 (3), 1966-1972). In this paper, we continue our study on finite groups (see Int. J. Theo. Physics, Group Theory, and Nonlinear Optics (2013), 17) and all the integer-valued characters of the above molecule are successfully derived.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"13 1","pages":"314-339"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627289","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}
We consider the reconstruction of shared secrets in communication networks, which are modelled by graphs whose components are subject to possible failure. The reconstruction probability can be approximated using minimal cuts, if the failure probabilities of vertices and edges are close to zero. As the main contribution of this paper, node separators are used to design a heuristic for the near-optimal placement of secrets sets on the vertices of the graph.
{"title":"Near-Optimal Placement of Secrets in Graphs","authors":"W. Poguntke","doi":"10.4236/OJDM.2016.64020","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64020","url":null,"abstract":"We consider the reconstruction of shared secrets in communication networks, which are modelled by graphs whose components are subject to possible failure. The reconstruction probability can be approximated using minimal cuts, if the failure probabilities of vertices and edges are close to zero. As the main contribution of this paper, node separators are used to design a heuristic for the near-optimal placement of secrets sets on the vertices of the graph.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"238-247"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626578","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}
Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles Ciin G such that s is maximum. In general, the maximum cycle packing problem is NP-hard. In this paper, it is shown for even graphs that if such a collection satisfies the condition that it minimizes the quantityon the set of all edge-disjoint cycle collections, then it is a maximum cycle packing. The paper shows that the determination of such a packing can be solved by a dynamic programming approach. For its solution, an-shortest path procedure on an appropriate acyclic networkis presented. It uses a particular monotonous node potential.
{"title":"A Dynamic Programming Approach for the Max-Min Cycle Packing Problem in Even Graphs","authors":"P. Recht","doi":"10.4236/OJDM.2016.64027","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64027","url":null,"abstract":"Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles Ciin G such that s is maximum. In general, the maximum cycle packing problem is NP-hard. In this paper, it is shown for even graphs that if such a collection satisfies the condition that it minimizes the quantityon the set of all edge-disjoint cycle collections, then it is a maximum cycle packing. The paper shows that the determination of such a packing can be solved by a dynamic programming approach. For its solution, an-shortest path procedure on an appropriate acyclic networkis presented. It uses a particular monotonous node potential.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"340-350"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626893","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}
The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.
{"title":"On Tilings of Quadrants and Rectangles and Rectangular Pattern","authors":"V. Nitica","doi":"10.4236/OJDM.2016.64028","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64028","url":null,"abstract":"The problem of tiling rectangles by polyominoes generated large interest. A related one is the problem of tiling parallelograms by twisted polyominoes. Both problems are related with tilings of (skewed) quadrants by polyominoes. Indeed, if all tilings of a (skewed) quadrant by a tile set can be reduced to a tiling by congruent rectangles (parallelograms), this provides information about tilings of rectangles (parallelograms). We consider a class of tile sets in a square lattice appearing from arbitrary dissections of rectangles in two L-shaped polyominoes and from symmetries of these tiles about the first bisector. Only translations of the tiles are allowed in a tiling. If the sides of the dissected rectangle are coprime, we show the existence of tilings of all (skewed) quadrants that do not follow the rectangular (parallelogram) pattern. If one of the sides of the dissected rectangle is 2 and the other is odd, we also show tilings of rectangles by the tile set that do not follow the rectangular pattern. If one of the sides of the dissected rectangle is 2 and the other side is even, we show a new infinite family of tile sets that follows the rectangular pattern when tiling one of the quadrants. For this type of dis-section, we also show a new infinite family that does not follow the rectangular pattern when tiling rectangles. Finally, we investigate more general dissections of rectangles, with. Here we show infinite families of tile sets that follow the rectangular pattern for a quadrant and infinite families that do not follow the rectangular pattern for any quadrant. We also show, for infinite families of tile sets of this type, tilings of rectangles that do not follow the rectangular pattern.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"351-371"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626906","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}
G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.
G. C. Ying, Y. Y.孟，B. E. Sagan, V. R. Vatter[1]找到了最多包含两个循环的连通图中最大独立集的最大个数。本文给出了n≥12阶连通图中最大独立集的最大数的一个替代证明。我们还描述了达到这个最大值的极值图。
{"title":"An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles","authors":"Min-Jen Jou, Jenq-Jong Lin","doi":"10.4236/OJDM.2016.64019","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64019","url":null,"abstract":"G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"227-237"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626523","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}
For a graph having no isolated vertex, a function is called an edge product cordial labeling of graph G, if the induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and the number of edges with label 1 differ by at most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at most 1. In this paper, we discuss edge product cordial labeling for some cycle related graphs.
{"title":"Edge Product Cordial Labeling of Some Cycle Related Graphs","authors":"U. M. Prajapati, N. B. Patel","doi":"10.4236/OJDM.2016.64023","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64023","url":null,"abstract":"For a graph having no isolated vertex, a function is called an edge product cordial labeling of graph G, if the induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and the number of edges with label 1 differ by at most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at most 1. In this paper, we discuss edge product cordial labeling for some cycle related graphs.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"268-278"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626660","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}
A graph is said to be a product cordial graph if there exists a function with each edge assign the label , such that the number of vertices with label 0 and the number of vertices with label 1 differ atmost by 1, and the number of edges with label 0 and the number of edges with label 1 differ by atmost 1. We discuss the product cordial labeling of the graphs obtained by duplication of some graph elements of gear graph. Also, we derive some product cordial graphs obtained by vertex switching operation on gear graph.
{"title":"Product Cordial Graph in the Context of Some Graph Operations on Gear Graph","authors":"U. M. Prajapati, K. K. Raval","doi":"10.4236/OJDM.2016.64022","DOIUrl":"https://doi.org/10.4236/OJDM.2016.64022","url":null,"abstract":"A graph is said to be a product cordial graph if there exists a function with each edge assign the label , such that the number of vertices with label 0 and the number of vertices with label 1 differ atmost by 1, and the number of edges with label 0 and the number of edges with label 1 differ by atmost 1. We discuss the product cordial labeling of the graphs obtained by duplication of some graph elements of gear graph. Also, we derive some product cordial graphs obtained by vertex switching operation on gear graph.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"259-267"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626644","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}
This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.
{"title":"Discrete Differential Geometry of Triangles and Escher-Style Trick Art","authors":"Naoto Morikawa","doi":"10.4236/OJDM.2016.63013","DOIUrl":"https://doi.org/10.4236/OJDM.2016.63013","url":null,"abstract":"This paper shows the usefulness of discrete differential geometry in global analysis. Using the discrete differential geometry of triangles, we could consider the global structure of closed trajectories (of triangles) on a triangular mesh consisting of congruent isosceles triangles. As an example, we perform global analysis of an Escher-style trick art, i.e., a simpler version of “Ascending and Descending”. After defining the local structure on the trick art, we analyze its global structure and attribute its paradox to a singular point (i.e., a singular triangle) at the center. Then, the endless “Penrose stairs” is described as a closed trajectory around the isolated singular point. The approach fits well with graphical projection and gives a simple and intuitive example of the interaction between global and local structures. We could deal with higher dimensional objects as well by considering n-simplices (n > 2) instead of triangles.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"06 1","pages":"161-166"},"PeriodicalIF":0.0,"publicationDate":"2016-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626754","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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