Let be a family of subgraphs of a graph G. An L-decomposition of G is an edge-disjoint decomposition of G into positive integer copies of Hi, where . Let Ck, Pk and Sk denote a cycle, a path and a star with k edges, respectively. For an integer , we prove that a balanced complete bipartite multigraph has a -decomposition if and only if k is even, and .
{"title":"{Ck, Pk, Sk} -Decompositions of Balanced Complete Bipartite Multigraphs","authors":"Jenq-Jong Lin, Min-Jen Jou","doi":"10.4236/OJDM.2016.63015","DOIUrl":"https://doi.org/10.4236/OJDM.2016.63015","url":null,"abstract":"Let be a family of subgraphs of a graph G. An L-decomposition of G is an edge-disjoint decomposition of G into positive integer copies of Hi, where . Let Ck, Pk and Sk denote a cycle, a path and a star with k edges, respectively. For an integer , we prove that a balanced complete bipartite multigraph has a -decomposition if and only if k is even, and .","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626842","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}
Both independence and independence-separation problems on chessboard graphs have been studied in detail, with hundreds of papers in the broader independence category, and several on the independence-separation problem variant for chessboard graphs. In this paper, the inde-pendence-separation problem is considered on the d-dimensional rook’s graph. A lower bound of k, for , is found for the independence-separation number on the d-dimensional rook’s graph, denoted by . For the case where , it is found that when n is odd and , . Conjecture and discussion are added.
{"title":"The Independence-Separation Problem on the 3-D Rook’s Graph","authors":"Paul A. Burchett","doi":"10.4236/OJDM.2016.63014","DOIUrl":"https://doi.org/10.4236/OJDM.2016.63014","url":null,"abstract":"Both independence and independence-separation problems on chessboard graphs have been studied in detail, with hundreds of papers in the broader independence category, and several on the independence-separation problem variant for chessboard graphs. In this paper, the inde-pendence-separation problem is considered on the d-dimensional rook’s graph. A lower bound of k, for , is found for the independence-separation number on the d-dimensional rook’s graph, denoted by . For the case where , it is found that when n is odd and , . Conjecture and discussion are added.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626796","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 proper edge t-coloring of a graph G is a coloring of its edges with colors 1, 2,..., t, such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each vertex, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. In this paper, we provide a new proof of the result on the colors in cyclically interval edge colorings of simple cycles which was first proved by Rafayel R. Kamalian in the paper “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles, Open Journal of Discrete Mathematics, 2013, 43-48”.
图G的正当边t染色是它的边的颜色为1,2，…， t，使得所有的颜色都被使用，并且没有两个相邻的边得到相同的颜色。图G的循环区间t-着色是G的适当边t-着色，使得对于每个顶点，在与x相关的边上使用的颜色集或在与x相关的边上不使用的颜色集形成整数区间。本文给出了由Rafayel R. Kamalian在“on a Number of colors In circular interval edge colorings of simple cycles”一文中首次证明的关于简单环的循环区间边着色的结果的一个新的证明，Open Journal of Discrete Mathematics, 2013, 43-48。
{"title":"Note on Cyclically Interval Edge Colorings of Simple Cycles","authors":"Nannan Wang, Yongqiang Zhao","doi":"10.4236/OJDM.2016.63016","DOIUrl":"https://doi.org/10.4236/OJDM.2016.63016","url":null,"abstract":"A proper edge t-coloring of a graph G is a coloring of its edges with colors 1, 2,..., t, such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each vertex, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. In this paper, we provide a new proof of the result on the colors in cyclically interval edge colorings of simple cycles which was first proved by Rafayel R. Kamalian in the paper “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles, Open Journal of Discrete Mathematics, 2013, 43-48”.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626441","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}
ErdOs asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1≤i≤n-1 ) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2≤n≤8 [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4 - 7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.
{"title":"On the ErdÖs Distance Conjecture in Geometry","authors":"A. Jafari, Amin Najafi Amin","doi":"10.4236/OJDM.2016.63012","DOIUrl":"https://doi.org/10.4236/OJDM.2016.63012","url":null,"abstract":"ErdOs asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1≤i≤n-1 ) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2≤n≤8 [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4 - 7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626817","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}
In this paper, we compute Atom-bond connectivity index, Fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, Geometric-arithmetic connectivity index and Fifth geometric-arithmetic connectivity index of Dutch windmill graph.
{"title":"Computation of Topological Indices of Dutch Windmill Graph","authors":"M. Kanna, R. P. Kumar, R. Jagadeesh","doi":"10.4236/OJDM.2016.62007","DOIUrl":"https://doi.org/10.4236/OJDM.2016.62007","url":null,"abstract":"In this paper, we compute Atom-bond connectivity index, Fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, Geometric-arithmetic connectivity index and Fifth geometric-arithmetic connectivity index of Dutch windmill graph.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626736","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}
In the present paper, geometry of the Boolean space Bn in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in Bn and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.
{"title":"Algebra and Geometry of Sets in Boolean Space","authors":"V. Leontiev, G. Movsisyan, Z. Margaryan","doi":"10.4236/OJDM.2016.62004","DOIUrl":"https://doi.org/10.4236/OJDM.2016.62004","url":null,"abstract":"In the \u0000present paper, geometry of the Boolean space Bn in terms of Hausdorff distances between subsets and \u0000subset sums is investigated. The main results are the algebraic and analytical \u0000expressions for representing of classical figures in Bn and the functions of distances between them. In \u0000particular, equations in sets are considered and their interpretations in \u0000combinatory terms are given.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626667","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}
In this paper, one introduces the polynomials Rn(x) and numbers Rn and derives some interesting identities related to the numbers and polynomials: Rn and Rn(x). We also give relation between the Stirling numbers, the Bell numbers, the Rn and Rn(x).
{"title":"On Polynomials Rn(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the p-Adic Integral on","authors":"H. Y. Lee, C. Ryoo","doi":"10.4236/OJDM.2016.62009","DOIUrl":"https://doi.org/10.4236/OJDM.2016.62009","url":null,"abstract":"In this paper, one introduces the polynomials Rn(x) and numbers Rn and derives some interesting identities related to the numbers and polynomials: Rn and Rn(x). We also give relation between the Stirling numbers, the Bell numbers, the Rn and Rn(x).","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626865","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 k-tree of a connected graph G is a spanning tree with maximum degree at most k. The rupture degree for a connected graph G is defined by , where and , respectively, denote the order of the largest component and number of components in . In this paper, we show that for a connected graph G, if for any cut-set , then G has a k-tree.
{"title":"The Rupture Degree of Graphs with k-Tree","authors":"Yinkui Li, Qingning Wang, Xiaolin Wang","doi":"10.4236/OJDM.2016.62011","DOIUrl":"https://doi.org/10.4236/OJDM.2016.62011","url":null,"abstract":"A k-tree of a connected graph G is a spanning tree with maximum degree \u0000at most k. The rupture degree for a \u0000connected graph G is defined by , where and , \u0000respectively, denote the order of the largest component and number of \u0000components in . In this \u0000paper, we show that for a connected graph G, \u0000if for any cut-set , then G has a k-tree.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626808","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 investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].
{"title":"On the Maximum Number of Dominating Classes in Graph Coloring","authors":"B. Zhou","doi":"10.4236/OJDM.2016.62006","DOIUrl":"https://doi.org/10.4236/OJDM.2016.62006","url":null,"abstract":"We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626722","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}
Pooya Daneshmand, Kamyar Mirzavaziri, M. Mirzavaziri
Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a permutation a is a double derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double derangements with respect to x and y. Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, we show that if and z is a permutation such that for and for , then where .
{"title":"Double Derangement Permutations","authors":"Pooya Daneshmand, Kamyar Mirzavaziri, M. Mirzavaziri","doi":"10.4236/OJDM.2016.62010","DOIUrl":"https://doi.org/10.4236/OJDM.2016.62010","url":null,"abstract":"Let n be a positive integer. A permutation a of the symmetric group of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a \u0000permutation a is a double \u0000derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double \u0000derangements with respect to x and y. \u0000Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result, \u0000we show that if and z is a permutation such that for and for , then where .","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626765","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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