The concept of linear tangle was introduced as an obstruction to mixed searching number. The concept of single ideal has been introduced as an obstruction to linear-width. Moreover, it was already known that mixed search number is equivalent to linear-width. Hence, by combining those results, we obtain a proof of the equivalence between linear tangle and single ideal. This short report gives an alternative proof of the equivalence.
{"title":"Equivalence between Linear Tangle and Maximal Single Ideal","authors":"T. Fujita, K. Yamazaki","doi":"10.4236/OJDM.2019.91002","DOIUrl":"https://doi.org/10.4236/OJDM.2019.91002","url":null,"abstract":"The concept of linear tangle was introduced as an obstruction to mixed searching number. The concept of single ideal has been introduced as an obstruction to linear-width. Moreover, it was already known that mixed search number is equivalent to linear-width. Hence, by combining those results, we obtain a proof of the equivalence between linear tangle and single ideal. This short report gives an alternative proof of the equivalence.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627242","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 n be a positive integer. For any integer a, we say that is idempotent modulo n if a2≡a(mod n). The n-modular Erdos-Burgess constant is the smallest positive integer l such that any l integers contain one or more integers, whose product is idempotent modulo n. We gave a sharp lower bound of the n-modular Erdos-Burgess constant, in particular, we determined the n-modular Erdos-Burgess constant in the case when n was a prime power or a product of pairwise distinct primes.
{"title":"On the Modular Erdös-Burgess Constant","authors":"Jun Hao, Haoli Wang, Lizhen Zhang","doi":"10.4236/OJDM.2019.91003","DOIUrl":"https://doi.org/10.4236/OJDM.2019.91003","url":null,"abstract":"Let n be a positive integer. For any integer a, we say that is idempotent modulo n if a2≡a(mod n). The n-modular Erdos-Burgess constant is the smallest positive integer l such that any l integers contain one or more integers, whose product is idempotent modulo n. We gave a sharp lower bound of the n-modular Erdos-Burgess constant, in particular, we determined the n-modular Erdos-Burgess constant in the case when n was a prime power or a product of pairwise distinct primes.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627321","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 note, for any pair of natural numbers (n,k), n≥3, k≥1, and 2k
对于任意一对自然数(n,k)， n≥3,k≥1,2k
{"title":"Infinite Parametric Families of Irreducible Polynomials with a Prescribed Number of Complex Roots","authors":"Catalin Nitica, V. Nitica","doi":"10.4236/ojdm.2019.91001","DOIUrl":"https://doi.org/10.4236/ojdm.2019.91001","url":null,"abstract":"In this note, for any pair of natural numbers (n,k), n≥3, k≥1, and 2k<n, we construct an infinite family of irreducible polynomials of degree n, with integer coefficients, that has exactly n-2k complex non-real roots if n is even and has exactly n-2k-1 complex non-real roots if n is odd. Our work generalizes a technical result of R. Bauer, presented in the classical monograph “Basic Algebra” of N. Jacobson. It is used there to construct polynomials with Galois groups, the symmetric group. Bauer’s result covers the case k=1 and n odd prime.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627224","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 : 2018-11-20DOI: 10.4236/ojdm.2023.132006
H. Najar, Riadh Gargouri
In this study we consider the problem of triangulated graphs. Precisely we give a necessary and sufficient condition for a graph to be triangulated. This give an alternative characterization of triangulated graphs. Our method is based on the so called perfectly nested sequences.
{"title":"A Remark on the Characterization of Triangulated Graphs","authors":"H. Najar, Riadh Gargouri","doi":"10.4236/ojdm.2023.132006","DOIUrl":"https://doi.org/10.4236/ojdm.2023.132006","url":null,"abstract":"In this study we consider the problem of triangulated graphs. Precisely we give a necessary and sufficient condition for a graph to be triangulated. This give an alternative characterization of triangulated graphs. Our method is based on the so called perfectly nested sequences.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44948228","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 uses the geometrical properties of L-convex polyominoes in order to reconstruct these polyominoes. The main idea is to modify some clauses to the original construction of Chrobak and Durr in order to control the L-convexity using 2SAT satisfaction problem.
{"title":"L -Convex Polyominoes: Discrete Tomographical Aspects","authors":"K. Tawbe, S. Mansour","doi":"10.4236/OJDM.2018.84009","DOIUrl":"https://doi.org/10.4236/OJDM.2018.84009","url":null,"abstract":"This paper uses the geometrical properties of L-convex polyominoes in order to reconstruct these polyominoes. The main idea is to modify some clauses to the original construction of Chrobak and Durr in order to control the L-convexity using 2SAT satisfaction problem.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"08 1","pages":"116-136"},"PeriodicalIF":0.0,"publicationDate":"2018-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42654993","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}
Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place in the network, determination of the structure of the network and community and description of the interactions between various elements of the network are the key issues in network analysis. One of the huge network structure challenges is the identification of the node(s) with an outstanding structural position within the network. The popular method for doing this is to calculate a measure of centrality. We examine node centrality measures such as degree, closeness, eigenvector, Katz and subgraph centrality for undirected networks. We show how the Katz centrality can be turned into degree and eigenvector centrality by considering limiting cases. Some existing centrality measures are linked to matrix functions. We extend this idea and examine the centrality measures based on general matrix functions and in particular, the logarithmic, cosine, sine, and hyperbolic functions. We also explore the concept of generalised Katz centrality. Various experiments are conducted for different networks generated by using random graph models. The results show that the logarithmic function in particular has potential as a centrality measure. Similar results were obtained for real-world networks.
{"title":"Centrality Measures Based on Matrix Functions","authors":"Lembris L Njotto","doi":"10.4236/ojdm.2018.84008","DOIUrl":"https://doi.org/10.4236/ojdm.2018.84008","url":null,"abstract":"Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place in the network, determination of the structure of the network and community and description of the interactions between various elements of the network are the key issues in network analysis. One of the huge network structure challenges is the identification of the node(s) with an outstanding structural position within the network. The popular method for doing this is to calculate a measure of centrality. We examine node centrality measures such as degree, closeness, eigenvector, Katz and subgraph centrality for undirected networks. We show how the Katz centrality can be turned into degree and eigenvector centrality by considering limiting cases. Some existing centrality measures are linked to matrix functions. We extend this idea and examine the centrality measures based on general matrix functions and in particular, the logarithmic, cosine, sine, and hyperbolic functions. We also explore the concept of generalised Katz centrality. Various experiments are conducted for different networks generated by using random graph models. The results show that the logarithmic function in particular has potential as a centrality measure. Similar results were obtained for real-world networks.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"08 1","pages":"79-115"},"PeriodicalIF":0.0,"publicationDate":"2018-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49270302","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 note we give a complete proof of a theorem of Dedekind.
本文给出了Dedekind定理的一个完整证明。
{"title":"On Tate’s Proof of a Theorem of Dedekind","authors":"Shiv Gupta","doi":"10.4236/OJDM.2018.83007","DOIUrl":"https://doi.org/10.4236/OJDM.2018.83007","url":null,"abstract":"In this note we give a complete proof of a theorem of Dedekind.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"08 1","pages":"73-78"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47637315","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 total coloring of a graph G with colors 1, 2, ..., t is called a cyclically interval total t-coloring if all colors are used, and the edges incident to each vertex v∈V(G) together with v are colored by (dG(v)+1) consecutive colors modulo t, where dG(v) is the degree of the vertex v in G. The one point union of k-copies of cycle Cn is the graph obtained by taking v as a common vertex such that any two distinct cycles and are edge disjoint and do not have any vertex in common except v. In this paper, we study the cyclically interval total colorings of , where n≥3 and k≥2.
{"title":"Cyclically Interval Total Coloring of the One Point Union of Cycles","authors":"Shijun Su, Wenwei Zhao, Yongqiang Zhao","doi":"10.4236/OJDM.2018.83006","DOIUrl":"https://doi.org/10.4236/OJDM.2018.83006","url":null,"abstract":"A total coloring of a graph G with colors 1, 2, ..., t is called a cyclically interval total t-coloring if all colors are used, and the edges incident to each vertex v∈V(G) together with v are colored by (dG(v)+1) consecutive colors modulo t, where dG(v) is the degree of the vertex v in G. The one point union of k-copies of cycle Cn is the graph obtained by taking v as a common vertex such that any two distinct cycles and are edge disjoint and do not have any vertex in common except v. In this paper, we study the cyclically interval total colorings of , where n≥3 and k≥2.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"08 1","pages":"65-72"},"PeriodicalIF":0.0,"publicationDate":"2018-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46443780","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 article, we establish some common fixed point results for two pairs of compatible mappings satisfying Meir-Keeler type contractive conditions in metric space and dislocated metric space which extend and improve some similar fixed point results in the literature.
{"title":"Some Common Fixed Point Theorems Satisfying Meir-Keeler Type Contractive Conditions","authors":"D. Panthi","doi":"10.4236/ojdm.2018.82004","DOIUrl":"https://doi.org/10.4236/ojdm.2018.82004","url":null,"abstract":"In this article, we establish some common fixed point results for two pairs of compatible mappings satisfying Meir-Keeler type contractive conditions in metric space and dislocated metric space which extend and improve some similar fixed point results in the literature.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"08 1","pages":"35-47"},"PeriodicalIF":0.0,"publicationDate":"2018-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48528008","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 a recent paper, we revisited Golomb’s hierarchy for tiling capabilities of finite sets of polyominoes. We considered the case when only translations are allowed for the tiles. In this classification, for several levels in Golomb’s hierarchy, more types appear. We showed that there is no general relationship among tiling capabilities for types corresponding to same level. Then we found the relationships from Golomb’s hierarchy that remain valid in this setup and found those that fail. As a consequence we discovered two alternative tiling hierarchies. The goal of this note is to study the validity of all implications in these new tiling hierarchies if one replaces the simply connected regions by deficient ones. We show that almost all of them fail. If one refines the hierarchy for tile sets that tile rectangles and for deficient regions then most of the implications of tiling capabilities can be recovered.
{"title":"Revisiting a Tiling Hierarchy (II)","authors":"V. Nitica","doi":"10.4236/OJDM.2018.82005","DOIUrl":"https://doi.org/10.4236/OJDM.2018.82005","url":null,"abstract":"In a recent paper, we revisited Golomb’s hierarchy for tiling capabilities of finite sets of polyominoes. We considered the case when only translations are allowed for the tiles. In this classification, for several levels in Golomb’s hierarchy, more types appear. We showed that there is no general relationship among tiling capabilities for types corresponding to same level. Then we found the relationships from Golomb’s hierarchy that remain valid in this setup and found those that fail. As a consequence we discovered two alternative tiling hierarchies. The goal of this note is to study the validity of all implications in these new tiling hierarchies if one replaces the simply connected regions by deficient ones. We show that almost all of them fail. If one refines the hierarchy for tile sets that tile rectangles and for deficient regions then most of the implications of tiling capabilities can be recovered.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":"08 1","pages":"48-63"},"PeriodicalIF":0.0,"publicationDate":"2018-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49553690","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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