Pub Date : 2021-01-01DOI: 10.4236/ojdm.2021.114009
Xia Hong, Guoyan Ao, F. Gao
{"title":"The Generalization of Signed Domination Number of Two Classes of Graphs","authors":"Xia Hong, Guoyan Ao, F. Gao","doi":"10.4236/ojdm.2021.114009","DOIUrl":"https://doi.org/10.4236/ojdm.2021.114009","url":null,"abstract":"","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627158","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 : 2021-01-01DOI: 10.4236/ojdm.2021.114008
Brice M. Miayoka, R. F. Babindamana, B. G. R. Bossoto
{"title":"Rational Points on Genus 3 Real Hyperelliptic Curves","authors":"Brice M. Miayoka, R. F. Babindamana, B. G. R. Bossoto","doi":"10.4236/ojdm.2021.114008","DOIUrl":"https://doi.org/10.4236/ojdm.2021.114008","url":null,"abstract":"","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70627124","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 : 2020-09-16DOI: 10.4236/ojdm.2020.104010
M. Hassan, Muhsin Al Hassan, Mazen Mostafa
Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : V(G)→{−1,1} if for every vertex v ∈ V(G), the sum of closed neighborhood weights of v is greater or equal to 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 6, 7 and arbitrary n.
{"title":"On Signed Domination of Grid Graph","authors":"M. Hassan, Muhsin Al Hassan, Mazen Mostafa","doi":"10.4236/ojdm.2020.104010","DOIUrl":"https://doi.org/10.4236/ojdm.2020.104010","url":null,"abstract":"Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : V(G)→{−1,1} if for every vertex v ∈ V(G), the sum of closed neighborhood weights of v is greater or equal to 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 6, 7 and arbitrary n.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47090494","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 : 2020-09-16DOI: 10.4236/OJDM.2020.104009
Pécs, Hungary
In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.
{"title":"Full-Rank Factoring of Elementary 2-Groups with Equal Size Factors","authors":"Pécs, Hungary","doi":"10.4236/OJDM.2020.104009","DOIUrl":"https://doi.org/10.4236/OJDM.2020.104009","url":null,"abstract":"In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41449271","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 : 2020-06-15DOI: 10.4236/ojdm.2020.103008
Charif Harrafa
In this article, we will present a particularly remarkable partitioning method of any infinite set with the aid of non-surjective injective maps. The non-surjective injective maps from an infinite set to itself constitute a semigroup for the law of composition bundled with certain properties allowing us to prove the existence of remarkable elements. Not to mention a compatible equivalence relation that allows transferring the said law to the quotient set, which can be provided with a lattice structure. Finally, we will present the concept of Co-injectivity and some of its properties.
{"title":"Partitioning of Any Infinite Set with the Aid of Non-Surjective Injective Maps and the Study of a Remarkable Semigroup","authors":"Charif Harrafa","doi":"10.4236/ojdm.2020.103008","DOIUrl":"https://doi.org/10.4236/ojdm.2020.103008","url":null,"abstract":"In this article, we will present a particularly remarkable partitioning method of any infinite set with the aid of non-surjective injective maps. The non-surjective injective maps from an infinite set to itself constitute a semigroup for the law of composition bundled with certain properties allowing us to prove the existence of remarkable elements. Not to mention a compatible equivalence relation that allows transferring the said law to the quotient set, which can be provided with a lattice structure. Finally, we will present the concept of Co-injectivity and some of its properties.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45155553","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 : 2020-01-22DOI: 10.4236/ojdm.2020.102005
M. Hassan, Muhsin Al Hassan, Mazen Mostafa
Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.
{"title":"The Signed Domination Number of Cartesian Product of Two Paths","authors":"M. Hassan, Muhsin Al Hassan, Mazen Mostafa","doi":"10.4236/ojdm.2020.102005","DOIUrl":"https://doi.org/10.4236/ojdm.2020.102005","url":null,"abstract":"Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43604913","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 : 2020-01-22DOI: 10.4236/ojdm.2020.102006
Isaac Reiter, Ju Zhou
Given a finite simple graph G, a set D ⊆ V(G) is called a dominating set if for all v ∈ V(G) , either v ∈ D or v is adjacent to some vertex in D. A dominating set D is independent if none of the vertices in D are adjacent, and D is perfect if each vertex not in D is adjacent to precisely one vertex in D. If a dominating set is both independent and perfect, then it is called an efficient dominating set. For a graph G, a set D is called a unique efficient dominating set of G if it is the only efficient dominating set of G. In this paper, the authors propose the definition of unique efficient dominating set, explore the properties of graphs with unique efficient dominating sets, and completely characterize several families of graphs which have unique efficient dominating sets.
{"title":"Unique Efficient Dominating Sets","authors":"Isaac Reiter, Ju Zhou","doi":"10.4236/ojdm.2020.102006","DOIUrl":"https://doi.org/10.4236/ojdm.2020.102006","url":null,"abstract":"Given a finite simple graph G, a set D ⊆ V(G) is called a dominating set if for all v ∈ V(G) , either v ∈ D or v is adjacent to some vertex in D. A dominating set D is independent if none of the vertices in D are adjacent, and D is perfect if each vertex not in D is adjacent to precisely one vertex in D. If a dominating set is both independent and perfect, then it is called an efficient dominating set. For a graph G, a set D is called a unique efficient dominating set of G if it is the only efficient dominating set of G. In this paper, the authors propose the definition of unique efficient dominating set, explore the properties of graphs with unique efficient dominating sets, and completely characterize several families of graphs which have unique efficient dominating sets.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44733853","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 : 2020-01-16DOI: 10.4236/ojdm.2020.101004
Paul A. Burchett
In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook’s graph. In a similar fashion, the roman domination number and the number of minimum roman dominating sets are found on the square bishop’s graph for odd board sizes. Also found are the number of minimum total dominating sets associated with the light-colored squares when n ≡ 1(mod12) (with n>1), and same for the dark-colored squares when n ≡ 7(mod12) .
{"title":"The Number of Minimum Roman and Minimum Total Dominating Sets for Some Chessboard Graphs","authors":"Paul A. Burchett","doi":"10.4236/ojdm.2020.101004","DOIUrl":"https://doi.org/10.4236/ojdm.2020.101004","url":null,"abstract":"In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook’s graph. In a similar fashion, the roman domination number and the number of minimum roman dominating sets are found on the square bishop’s graph for odd board sizes. Also found are the number of minimum total dominating sets associated with the light-colored squares when n ≡ 1(mod12) (with n>1), and same for the dark-colored squares when n ≡ 7(mod12) .","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45256526","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 : 2020-01-01DOI: 10.4236/ojdm.2020.101001
Catalin Nitica, V. Nitica
Let b ≥ 2 be a numeration base. A b-weak additive Ramanujan-Hardy (or b-wARH) number N is a non-negative integer for which there exists at least one non-negative integer A, such that the sum of A and the sum of base b digits of N, added to the reversal of the sum, give N. We say that a pair of such numbers are related of degrees d ≥ 0 if their difference is d. We show for all numeration bases an infinity of degrees d for which there exists an infinity of pairs of b-wARH numbers related of degree d.
{"title":"Infinite Sets of Related b-wARH Pairs","authors":"Catalin Nitica, V. Nitica","doi":"10.4236/ojdm.2020.101001","DOIUrl":"https://doi.org/10.4236/ojdm.2020.101001","url":null,"abstract":"Let b ≥ 2 be a numeration base. A b-weak additive Ramanujan-Hardy (or b-wARH) number N is a non-negative integer for which there exists at least one non-negative integer A, such that the sum of A and the sum of base b digits of N, added to the reversal of the sum, give N. We say that a pair of such numbers are related of degrees d ≥ 0 if their difference is d. We show for all numeration bases an infinity of degrees d for which there exists an infinity of pairs of b-wARH numbers related of degree d.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626916","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 : 2020-01-01DOI: 10.4236/ojdm.2020.101002
M. M. Al-Shamiri, S. Nada, A. Elrokh, Yasser Elmshtaye
A digraph is a graph in which each edge has an orientation. A linear directed path, , is a path whose all edges have the same orientation. A linear simple graph is called directed cordial if it admits 0 - 1 labeling that satisfies certain condition. In this paper, we study the cordiality of directed paths and their second power . Similar studies are done for and the join . We show that , and are directed cordial. Sufficient conditions are given to the join to be directed cordial.
{"title":"Some Results on Cordial Digraphs","authors":"M. M. Al-Shamiri, S. Nada, A. Elrokh, Yasser Elmshtaye","doi":"10.4236/ojdm.2020.101002","DOIUrl":"https://doi.org/10.4236/ojdm.2020.101002","url":null,"abstract":"A digraph is a graph in which each edge has an orientation. A linear directed path, , is a path whose all edges have the same orientation. A linear simple graph is called directed cordial if it admits 0 - 1 labeling that satisfies certain condition. In this paper, we study the cordiality of directed paths and their second power . Similar studies are done for and the join . We show that , and are directed cordial. Sufficient conditions are given to the join to be directed cordial.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70626931","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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