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Size Ramsey number of bipartite graphs and bipartite Ramanujan graphs 二部图和二部Ramanujan图的Size-Ramsey数
IF 0.4
Transactions on Combinatorics Pub Date : 2019-06-01 DOI: 10.22108/TOC.2019.111317.1573
R. Javadi, Farideh Khoeini
{"title":"Size Ramsey number of bipartite graphs and bipartite Ramanujan graphs","authors":"R. Javadi, Farideh Khoeini","doi":"10.22108/TOC.2019.111317.1573","DOIUrl":"https://doi.org/10.22108/TOC.2019.111317.1573","url":null,"abstract":"Given a graph $ G $, a graph $ F $ is said to be Ramsey for $ G $ if in every edge coloring of $F$ with two colors, there exists a monochromatic copy of $G$. The minimum number of edges of a graph $ F $ which is Ramsey for $ G $ is called the size-Ramsey number of $G$ and is denoted by $hat{r}(G)$. In 1983, Beck gave a linear upper bound (in terms of $n$) for $hat{r}(P_{n})$, where $ P_n $ is a path on $ n $ vertices, giving a positive answer to a question of ErdH{o}s. After that, different approaches were attempted by several authors to reduce the upper bound for $hat{r}(P_n)$ for sufficiently large $n$ and most of these approaches are based on the classic models of random graphs. Also, Haxell and Kohayakama in 1994 proved that the size Ramsey number of the cycle $ C_n $ is linear in terms $ n $, however the Szemeredi's regularity lemma is used in their proof and so no specific constant coefficient is provided. Here, we provide a method to obtain an upper bound for the size Ramsey number of a graph using good expander graphs such as Ramanujan graphs. In particular, we give an alternative proof for the linearity of the size Ramsey number of paths and cycles. Our method has two privileges in compare to the previous ones. Firstly, it proves the upper bound for every positive integer $n$ in comparison to the random graph methods which needs $ n $ to be sufficiently large. Also, due to the recent explicit constructions for bipartite Ramanujan graphs by Marcus, Spielman and Srivastava, we can constructively find the graphs with small sizes which are Ramsey for a given graph. We also obtain some results about the bipartite Ramsey numbers.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"45-51"},"PeriodicalIF":0.4,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43234001","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 derivable trees 关于可导树
IF 0.4
Transactions on Combinatorics Pub Date : 2019-06-01 DOI: 10.22108/TOC.2019.113737.1601
M. Hamidi, A. Saeid
{"title":"On derivable trees","authors":"M. Hamidi, A. Saeid","doi":"10.22108/TOC.2019.113737.1601","DOIUrl":"https://doi.org/10.22108/TOC.2019.113737.1601","url":null,"abstract":"This paper defines the concept of partitioned hypergraphs‎, ‎and enumerates the number of these hypergraphs and discrete complete hypergraphs‎. ‎A positive equivalence relation is defined on hypergraphs‎, ‎this relation establishes a connection between hypergraphs and graphs‎. ‎Moreover‎, ‎we define the concept of (extended) derivable graph‎. ‎Then a connection between hypergraphs and (extended) derivable graphs was investigated‎. ‎Via the positive equivalence relation on hypergraphs‎, ‎we show that some special trees are derivable graph and complete graphs are self derivable graphs‎.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"21-43"},"PeriodicalIF":0.4,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47035412","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: 7
A note on some lower bounds of the Laplacian energy of a graph 关于图的拉普拉斯能量的一些下界的注释
IF 0.4
Transactions on Combinatorics Pub Date : 2019-06-01 DOI: 10.22108/TOC.2019.115269.1616
I. Milovanovic, M. Matejic, P. Milošević, E. Milovanovic, Akbar Ali
{"title":"A note on some lower bounds of the Laplacian energy of a graph","authors":"I. Milovanovic, M. Matejic, P. Milošević, E. Milovanovic, Akbar Ali","doi":"10.22108/TOC.2019.115269.1616","DOIUrl":"https://doi.org/10.22108/TOC.2019.115269.1616","url":null,"abstract":"‎‎‎For a simple connected graph $G$ of order $n$ and size $m$‎, ‎the Laplacian energy of $G$ is defined as‎ ‎$LE(G)=sum_{i=1}^n|mu_i-frac{2m}{n}|$ where $mu_1‎, ‎mu_2,ldots‎,‎‎mu_{n-1}‎, ‎mu_{n}$‎ ‎are the Laplacian eigenvalues of $G$ satisfying $mu_1ge mu_2gecdots ge mu_{n-1}>‎ ‎mu_{n}=0$‎. ‎In this note‎, ‎some new lower bounds on the graph invariant $LE(G)$ are derived‎. ‎The obtained results are compared with some already known lower bounds of $LE(G)$‎.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"13-19"},"PeriodicalIF":0.4,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43819473","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
Bounds for the skew Laplacian (skew adjacency) spectral radius of a digraph 有向图的斜拉普拉斯(斜邻接)谱半径的边界
IF 0.4
Transactions on Combinatorics Pub Date : 2019-03-06 DOI: 10.22108/TOC.2019.112589.1582
H. A. Ganie
{"title":"Bounds for the skew Laplacian (skew adjacency) spectral radius of a digraph","authors":"H. A. Ganie","doi":"10.22108/TOC.2019.112589.1582","DOIUrl":"https://doi.org/10.22108/TOC.2019.112589.1582","url":null,"abstract":"‎‎For a simple connected graph $G$ with $n$ vertices and $m$ edges‎, ‎let $overrightarrow{G}$ be a digraph obtained by giving an arbitrary direction to the edges of $G$‎. ‎In this paper‎, ‎we consider the skew Laplacian/skew adjacency matrix of the digraph $overrightarrow{G}$‎. ‎We obtain upper bounds for the skew Laplacian/skew adjacency spectral radius‎, ‎in terms of various parameters (like oriented degree‎, ‎average oriented degree) associated with the structure of the digraph $overrightarrow{G}$‎. ‎We also obtain upper and lower bounds for the skew Laplacian/skew adjacency spectral radius‎, ‎in terms of skew Laplacian/skew adjacency rank of the digraph $overrightarrow{G}$‎.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"1-12"},"PeriodicalIF":0.4,"publicationDate":"2019-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44810217","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: 7
On the double bondage number of graphs products 关于图乘积的二重束缚数
IF 0.4
Transactions on Combinatorics Pub Date : 2019-03-01 DOI: 10.22108/TOC.2018.114111.1605
H. Maimani, Z. Koushki
{"title":"On the double bondage number of graphs products","authors":"H. Maimani, Z. Koushki","doi":"10.22108/TOC.2018.114111.1605","DOIUrl":"https://doi.org/10.22108/TOC.2018.114111.1605","url":null,"abstract":"A set $D$ of vertices of graph $G$ is called $double$ $dominating$ $set$ if for any vertex $v$, $|N[v]cap D|geq 2$. The minimum cardinality of $double$ $domination$ of $G$ is denoted by $gamma_d(G)$. The minimum number of edges $E'$ such that $gamma_d(Gsetminus E)>gamma_d(G)$ is called the double bondage number of $G$ and is denoted by $b_d(G)$. This paper determines that $b_d(Gvee H)$ and exact values of $b(P_ntimes P_2)$, and generalized corona product of graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"51-59"},"PeriodicalIF":0.4,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47519518","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 the zero forcing number of generalized Sierpinski graphs 关于广义Sierpinski图的迫零数
IF 0.4
Transactions on Combinatorics Pub Date : 2019-03-01 DOI: 10.22108/TOC.2018.101107.1463
E. Vatandoost, F. Ramezani, S. Alikhani
{"title":"On the zero forcing number of generalized Sierpinski graphs","authors":"E. Vatandoost, F. Ramezani, S. Alikhani","doi":"10.22108/TOC.2018.101107.1463","DOIUrl":"https://doi.org/10.22108/TOC.2018.101107.1463","url":null,"abstract":"In this article we study the Zero forcing number of Generalized Sierpi'{n}ski graphs $S(G,t)$‎. ‎More precisely‎, ‎we obtain a general lower bound on the Zero forcing number of $S(G,t)$ and we show that this bound is tight‎. ‎In particular‎, ‎we consider the cases in which the base graph $G$ is a star‎, ‎path‎, ‎a cycle or a complete graph‎.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"41-50"},"PeriodicalIF":0.4,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45859856","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
On the defensive alliances in graph 论图中的防御联盟
IF 0.4
Transactions on Combinatorics Pub Date : 2019-03-01 DOI: 10.22108/TOC.2018.50156.1396
H. Kharazi, A. Tehrani
{"title":"On the defensive alliances in graph","authors":"H. Kharazi, A. Tehrani","doi":"10.22108/TOC.2018.50156.1396","DOIUrl":"https://doi.org/10.22108/TOC.2018.50156.1396","url":null,"abstract":"‎Let $ G = (V,E) $ be a graph‎. ‎We say that $ S subseteq V $ is a defensive alliance if for every $ u in S $‎, ‎the number of neighbors $ u $ has in $ S $ plus one (counting $ u $) is at least as large as the number of neighbors it has outside $ S $‎. ‎Then‎, ‎for every vertex $ u $ in a defensive alliance $ S $‎, ‎any attack on a single vertex by the neighbors of $ u $ in $ V-S $ can be thwarted by the neighbors of $ u $ in $ S $ and $ u $ itself‎. ‎In this paper‎, ‎we study alliances that are containing a given vertex $ u $ and study their mathematical properties‎.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"1-14"},"PeriodicalIF":0.4,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44525722","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
A generalization of global dominating function 全局支配函数的一种推广
IF 0.4
Transactions on Combinatorics Pub Date : 2019-03-01 DOI: 10.22108/TOC.2019.110404.1562
Mostafa Momeni, A. Zaeembashi
{"title":"A generalization of global dominating function","authors":"Mostafa Momeni, A. Zaeembashi","doi":"10.22108/TOC.2019.110404.1562","DOIUrl":"https://doi.org/10.22108/TOC.2019.110404.1562","url":null,"abstract":"Let G be a graph. A function f : V (G) −→ {0, 1}, satisfying the condition that every vertex u with f(u) = 0 is adjacent with at least one vertex v such that f(v) = 1, is called a dominating function (DF ). The weight of f is defined as wet(f) = Σv∈V (G)f(v). The minimum weight of a dominating function of G is denoted by γ(G), and is called the domination number of G. A dominating function f is called a global dominating function (GDF ) if f is also a DF of G. The minimum weight of a global dominating function is denoted by γg(G) and is called global domination number of G. In this paper we introduce a generalization of global dominating function. Suppose G is a graph and s ≥ 2 and Kn is the complete graph on V (G). A function f : V (G) −→ {0, 1} on G is s-dominating function (s−DF ), if there exists some factorization {G1, . . . , Gs} of Kn, such that G1 = G and f is dominating function of each Gi.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"61-68"},"PeriodicalIF":0.4,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47145636","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 problems concerning fixed-point-free permutations and on the polycirculant conjecture-a survey 关于无不动点置换问题和多循环猜想的综述
IF 0.4
Transactions on Combinatorics Pub Date : 2019-03-01 DOI: 10.22108/TOC.2018.112665.1585
M. Arezoomand, A. Abdollahi, Pablo Spiga
{"title":"On problems concerning fixed-point-free permutations and on the polycirculant conjecture-a survey","authors":"M. Arezoomand, A. Abdollahi, Pablo Spiga","doi":"10.22108/TOC.2018.112665.1585","DOIUrl":"https://doi.org/10.22108/TOC.2018.112665.1585","url":null,"abstract":"Fixed-point-free permutations‎, ‎also known as derangements‎, ‎have been studied for centuries‎. ‎In particular‎, ‎depending on their applications‎, ‎derangements of prime-power order and of prime order have always played a crucial role in a variety of different branches of mathematics‎: ‎from number theory to algebraic graph theory‎. ‎Substantial progress has been made on the study of derangements‎, ‎many long-standing open problems have been solved‎, ‎and many new research problems have arisen‎. ‎The results obtained and the methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs‎. ‎The methods used in this area range from deep group theory‎, ‎including the classification of the finite simple groups‎, ‎to combinatorial techniques‎. ‎This article is devoted to surveying results‎, ‎open problems and methods in this area‎.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"15-40"},"PeriodicalIF":0.4,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44084540","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: 9
Visual cryptography scheme on graphs with $m^{*}(G)=4$ $m^｛*｝（G）=4图的可视化密码方案$
IF 0.4
Transactions on Combinatorics Pub Date : 2019-02-04 DOI: 10.22108/TOC.2019.113671.1599
M. Davarzani
{"title":"Visual cryptography scheme on graphs with $m^{*}(G)=4$","authors":"M. Davarzani","doi":"10.22108/TOC.2019.113671.1599","DOIUrl":"https://doi.org/10.22108/TOC.2019.113671.1599","url":null,"abstract":"‎Let $G=(V,E)$ be a connected graph and $Gamma (G)$ be the strong access structure where obtained from graph $G$‎. ‎A visual cryptography scheme (VCS) for a set $P$ of participants is a method to encode a secret image such that any pixel of this image change to $m$ subpixels and only qualified sets can recover the secret image by stacking their shares‎. ‎The value of $m$ is called the pixel expansion and the minimum value of the pixel expansion of a VCS for $Gamma (G)$ is denoted by $m^{*}(G)$‎. ‎In this paper we obtain a characterization of all connected graphs $G$ with $m^{*}(G)=4$ and $omega (G)=5$ which $omega(G)$ is the clique number of graph $G$‎.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"8 1","pages":"53-66"},"PeriodicalIF":0.4,"publicationDate":"2019-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43981176","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
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