Pub Date : 2019-06-01DOI: 10.22108/TOC.2019.111317.1573
R. Javadi, Farideh Khoeini
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.
给定一个图$G$，如果在$F$的每一个有两种颜色的边着色中，存在$G$的一个单色副本，则称$F$是$G$的Ramsey。图$ F $的最小边数是$G$的拉姆齐数，称为$G$的大小拉姆齐数，用$hat{r}(G)$表示。1983年，Beck给出了$hat{r}(P_{n})$的线性上界(以$n$表示)，其中$ P_n $是$n$个顶点上的路径，给出了ErdH{o}s问题的正答案。之后，一些作者尝试了不同的方法来降低$hat{r}(P_n)$的上界，对于足够大的$n$，这些方法大多是基于经典的随机图模型。1994年Haxell和Kohayakama也证明了循环C_n $的大小Ramsey数在n $项上是线性的，但是在他们的证明中使用了Szemeredi的正则性引理，因此没有给出具体的常系数。在这里，我们提供了一种方法来获得一个图的大小拉姆齐数的上界使用良好的展开图，如拉马努金图。特别地，我们给出了路径和环的拉姆齐数大小的线性的另一种证明。与前面的方法相比，我们的方法有两个特权。首先，与随机图方法相比，它证明了每个正整数$n$的上界，而随机图方法需要$n$足够大。此外，由于Marcus, Spielman和Srivastava最近对二部Ramanujan图的显式构造，我们可以建设性地找到给定图的Ramsey小尺寸图。我们还得到了关于二部拉姆齐数的一些结果。
{"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}
Pub Date : 2019-06-01DOI: 10.22108/TOC.2019.113737.1601
M. Hamidi, A. Saeid
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.
{"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}
Pub Date : 2019-06-01DOI: 10.22108/TOC.2019.115269.1616
I. Milovanovic, M. Matejic, P. Milošević, E. Milovanovic, Akbar Ali
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 $LE(G)$.
{"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}
Pub Date : 2019-03-06DOI: 10.22108/TOC.2019.112589.1582
H. A. Ganie
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 $overrightarrow{G}$.
{"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}
Pub Date : 2019-03-01DOI: 10.22108/TOC.2018.114111.1605
H. Maimani, Z. Koushki
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.
{"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}
Pub Date : 2019-03-01DOI: 10.22108/TOC.2018.101107.1463
E. Vatandoost, F. Ramezani, S. Alikhani
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.
{"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}
Pub Date : 2019-03-01DOI: 10.22108/TOC.2018.50156.1396
H. Kharazi, A. Tehrani
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.
设$ G = (V,E) $是一个图。我们说S subseteq V是一个防守联盟如果美元每u S美元,邻居u美元的数量在S + 1美元(u)美元计算的数量至少是一样大邻居年代美元以外。然后，对于防御联盟S $中的每个顶点$ u $， $ V-S $中$ u $的邻居对单个顶点的任何攻击都可以被$ S $和$ u $自身中的$ u $的邻居所挫败。在本文中，我们研究了包含给定顶点u的联盟，并研究了它们的数学性质。
{"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}
Pub Date : 2019-03-01DOI: 10.22108/TOC.2019.110404.1562
Mostafa Momeni, A. Zaeembashi
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.
{"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}
Pub Date : 2019-03-01DOI: 10.22108/TOC.2018.112665.1585
M. Arezoomand, A. Abdollahi, Pablo Spiga
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.
{"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}
Pub Date : 2019-02-04DOI: 10.22108/TOC.2019.113671.1599
M. Davarzani
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 $G$.
{"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}
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