Pub Date : 2020-12-01DOI: 10.22108/TOC.2020.116355.1633
A. Khamseh
Let $chi_{gl}(G)$ be the {it{group choice number}} of $G$. A graph $G$ is called {it{edge-$k$-group choosable}} if its line graph is $k$-group choosable. The {it{group-choice index}} of $G$, $chi'_{gl}(G)$, is the smallest $k$ such that $G$ is edge-$k$-group choosable, that is, $chi'_{gl}(G)$ is the group chice number of the line graph of $G$, $chi_{gl}(ell(G))$. It is proved that, if $G$ is an outerplanar graph with maximum degree $D<5$, or if $G$ is a $({K_2}^c+(K_1 cup K_2))$-minor-free graph, then $chi'_{gl}(G)leq D(G)+1$. As a straightforward consequence, every $K_{2,3}$-minor-free graph $G$ or every $K_4$-minor-free graph $G$ is edge-$(D(G)+1)$-group choosable. Moreover, it is proved that if $G$ is an outerplanar graph with maximum degree $Dgeq 5$, then $chi'_{gl}(G)leq D$.
{"title":"Edge-group choosability of outerplanar and near-outerplanar graphs","authors":"A. Khamseh","doi":"10.22108/TOC.2020.116355.1633","DOIUrl":"https://doi.org/10.22108/TOC.2020.116355.1633","url":null,"abstract":"Let $chi_{gl}(G)$ be the {it{group choice number}} of $G$. A graph $G$ is called {it{edge-$k$-group choosable}} if its line graph is $k$-group choosable. The {it{group-choice index}} of $G$, $chi'_{gl}(G)$, is the smallest $k$ such that $G$ is edge-$k$-group choosable, that is, $chi'_{gl}(G)$ is the group chice number of the line graph of $G$, $chi_{gl}(ell(G))$. It is proved that, if $G$ is an outerplanar graph with maximum degree $D<5$, or if $G$ is a $({K_2}^c+(K_1 cup K_2))$-minor-free graph, then $chi'_{gl}(G)leq D(G)+1$. As a straightforward consequence, every $K_{2,3}$-minor-free graph $G$ or every $K_4$-minor-free graph $G$ is edge-$(D(G)+1)$-group choosable. Moreover, it is proved that if $G$ is an outerplanar graph with maximum degree $Dgeq 5$, then $chi'_{gl}(G)leq D$.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"211-216"},"PeriodicalIF":0.4,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68209197","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 : 2020-09-01DOI: 10.22108/TOC.2020.120019.1685
Serkan Kader, B. Ö. Güler, E. Akşit
n this paper, we investigate suborbital graphs formed by $Nbig(Gamma_0(N)big)$-invariant equivalence relation induced on $hat{mathbb{Q}}$. Conditions for being an edge are obtained as a main tool, then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated.
{"title":"On quadrilaterals in the suborbital graphs of the normalizer","authors":"Serkan Kader, B. Ö. Güler, E. Akşit","doi":"10.22108/TOC.2020.120019.1685","DOIUrl":"https://doi.org/10.22108/TOC.2020.120019.1685","url":null,"abstract":"n this paper, we investigate suborbital graphs formed by $Nbig(Gamma_0(N)big)$-invariant equivalence relation induced on $hat{mathbb{Q}}$. Conditions for being an edge are obtained as a main tool, then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"147-159"},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47311883","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 : 2020-09-01DOI: 10.22108/TOC.2020.121874.1713
R. Kazemi, A. Behtoei
The Hosoya index, also known as the $Z$ index, of a graph is the total number of matchings in it. In this paper, we study the Hosoya index of the tree structures. Our aim is to give some results on $Z$ in terms of Fibonacci numbers in such structures. Also, the asymptotic normality of this index is given.
{"title":"Hosoya index of tree structures","authors":"R. Kazemi, A. Behtoei","doi":"10.22108/TOC.2020.121874.1713","DOIUrl":"https://doi.org/10.22108/TOC.2020.121874.1713","url":null,"abstract":"The Hosoya index, also known as the $Z$ index, of a graph is the total number of matchings in it. In this paper, we study the Hosoya index of the tree structures. Our aim is to give some results on $Z$ in terms of Fibonacci numbers in such structures. Also, the asymptotic normality of this index is given.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"161-169"},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43414432","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 : 2020-09-01DOI: 10.22108/TOC.2020.116372.1634
Zikai Tang, Renfang Wu, Hanlin Chen, H. Deng
Let $G$ be a connected graph with vertex set $V(G)={v_1, v_2,ldots,v_n}$. The distance matrix $D=D(G)$ of $G$ is defined so that its $(i,j)$-entry is equal to the distance $d_G(v_i,v_j)$ between the vertices $v_i$ and $v_j$ of $G$. The eigenvalues ${mu_1, mu_2,ldots,mu_n}$ of $D(G)$ are the $D$-eigenvalues of $G$ and form the distance spectrum or the $D$-spectrum of $G$, denoted by $Spec_D(G)$. In this paper, we introduce two new operations $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ on graphs $G_1$ and $G_2$, and describe the distance spectra of $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ of regular graphs $G_1$ and $G_2 $ in terms of their adjacency spectra. By using these results, we obtain some new integral adjacency spectrum graphs, integral distance spectrum graphs and a number of families of sets of noncospectral graphs with equal distance energy.
{"title":"The distance spectrum of two new operations of graphs","authors":"Zikai Tang, Renfang Wu, Hanlin Chen, H. Deng","doi":"10.22108/TOC.2020.116372.1634","DOIUrl":"https://doi.org/10.22108/TOC.2020.116372.1634","url":null,"abstract":"Let $G$ be a connected graph with vertex set $V(G)={v_1, v_2,ldots,v_n}$. The distance matrix $D=D(G)$ of $G$ is defined so that its $(i,j)$-entry is equal to the distance $d_G(v_i,v_j)$ between the vertices $v_i$ and $v_j$ of $G$. The eigenvalues ${mu_1, mu_2,ldots,mu_n}$ of $D(G)$ are the $D$-eigenvalues of $G$ and form the distance spectrum or the $D$-spectrum of $G$, denoted by $Spec_D(G)$. In this paper, we introduce two new operations $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ on graphs $G_1$ and $G_2$, and describe the distance spectra of $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ of regular graphs $G_1$ and $G_2 $ in terms of their adjacency spectra. By using these results, we obtain some new integral adjacency spectrum graphs, integral distance spectrum graphs and a number of families of sets of noncospectral graphs with equal distance energy.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"125-138"},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44361085","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 : 2020-09-01DOI: 10.22108/TOC.2020.119553.1680
H. T. Faal
In this paper, we first extend the weighted handshaking lemma, using a generalization of the concept of the degree of vertices to the values of graphs. This edge-version of the weighted handshaking lemma yields an immediate generalization of the Mantel's classical result which asks for the maximum number of edges in triangle-free graphs to the class of $K_{4}$-free graphs. Then, by defining the concept of value for cliques (complete subgraphs) of higher orders, we also extend the classical result of Mantel for any graph $G$. We finally conclude our paper with a discussion about the possible future works.
{"title":"On clique values identities and mantel-type theorems","authors":"H. T. Faal","doi":"10.22108/TOC.2020.119553.1680","DOIUrl":"https://doi.org/10.22108/TOC.2020.119553.1680","url":null,"abstract":"In this paper, we first extend the weighted handshaking lemma, using a generalization of the concept of the degree of vertices to the values of graphs. This edge-version of the weighted handshaking lemma yields an immediate generalization of the Mantel's classical result which asks for the maximum number of edges in triangle-free graphs to the class of $K_{4}$-free graphs. Then, by defining the concept of value for cliques (complete subgraphs) of higher orders, we also extend the classical result of Mantel for any graph $G$. We finally conclude our paper with a discussion about the possible future works.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"139-146"},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45114499","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 : 2020-05-03DOI: 10.22108/TOC.2020.120320.1692
T. C. Bruen
Consider a v × v (0, 1) matrix A with exactly k ones in each row and each column. A is (λ, n)–stable, if it does not contain any λ × n submatrix with exactly one 0. If A is (λ, n)–stable, λ, n ≥ 2, then under suitable conditions on A, v ≥ k k(n−1)+(λ−2) . The case n λ−2 of equality leads to new and substantive connections with block designs. The previous bound and characterization of (λ, 2)–stable matrices follows immediately as a special case.
{"title":"Exact bounds for (λ,n)–stable 0-1 matrices.","authors":"T. C. Bruen","doi":"10.22108/TOC.2020.120320.1692","DOIUrl":"https://doi.org/10.22108/TOC.2020.120320.1692","url":null,"abstract":"Consider a v × v (0, 1) matrix A with exactly k ones in each row and each column. A is (λ, n)–stable, if it does not contain any λ × n submatrix with exactly one 0. If A is (λ, n)–stable, λ, n ≥ 2, then under suitable conditions on A, v ≥ k k(n−1)+(λ−2) . The case n λ−2 of equality leads to new and substantive connections with block designs. The previous bound and characterization of (λ, 2)–stable matrices follows immediately as a special case.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"171-180"},"PeriodicalIF":0.4,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48979402","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 : 2020-04-30DOI: 10.22108/TOC.2020.120014.1684
H. A. Ahangar
A 2-rainbow dominating function (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin V(G)$ with $f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$ is fulfilled, where $N(v)$ is the open neighborhood of $v$. A maximal 2-rainbow dominating function of a graph $G$ is a $2$-rainbow dominating function $f$ such that the set ${winV(G)|f(w)=emptyset}$ is not a dominating set of $G$. The weight of a maximal 2RDF $f$ is the value $omega(f)=sum_{vin V}|f (v)|$. The maximal $2$-rainbow domination number of a graph $G$, denoted by $gamma_{m2r}(G)$, is the minimum weight of a maximal 2RDF of $G$. In this paper, we continue the study of maximal 2-rainbow domination {number} in graphs. Specially, we first characterize all graphs with large maximal 2-rainbow domination number. Finally, we determine the maximal $2$-rainbow domination number in the sun and sunlet graphs.
{"title":"Further results on maximal rainbow domination number","authors":"H. A. Ahangar","doi":"10.22108/TOC.2020.120014.1684","DOIUrl":"https://doi.org/10.22108/TOC.2020.120014.1684","url":null,"abstract":"A 2-rainbow dominating function (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin V(G)$ with $f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$ is fulfilled, where $N(v)$ is the open neighborhood of $v$. A maximal 2-rainbow dominating function of a graph $G$ is a $2$-rainbow dominating function $f$ such that the set ${winV(G)|f(w)=emptyset}$ is not a dominating set of $G$. The weight of a maximal 2RDF $f$ is the value $omega(f)=sum_{vin V}|f (v)|$. The maximal $2$-rainbow domination number of a graph $G$, denoted by $gamma_{m2r}(G)$, is the minimum weight of a maximal 2RDF of $G$. In this paper, we continue the study of maximal 2-rainbow domination {number} in graphs. Specially, we first characterize all graphs with large maximal 2-rainbow domination number. Finally, we determine the maximal $2$-rainbow domination number in the sun and sunlet graphs.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"201-210"},"PeriodicalIF":0.4,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41681563","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 : 2020-04-30DOI: 10.22108/TOC.2020.116191.1628
J. John
Let x be a vertex of a connected graph G and W ⊂V(G) such that x∉W.Then W is called an x - Steiner set of G if W⋃{x} is a steiner set of G. The minimum cardinality of an x - Steiner set of G is defined as x - Steiner number of G and denoted by s_x (G). Some general properties satisfied by this concept are studied. The x - Steiner numbers of certain classes of graphs are determined. Connected graphs of order p with x - Steiner number 1 or p-1 are characterized. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G such that s(G) = a and s_x (G)= b for some vertex x in G, where s(G) is the Steiner number of a graph.
{"title":"The Vertex Steiner Number of a Graph","authors":"J. John","doi":"10.22108/TOC.2020.116191.1628","DOIUrl":"https://doi.org/10.22108/TOC.2020.116191.1628","url":null,"abstract":"Let x be a vertex of a connected graph G and W ⊂V(G) such that x∉W.Then W is called an x - Steiner set of G if W⋃{x} is a steiner set of G. The minimum cardinality of an x - Steiner set of G is defined as x - Steiner number of G and denoted by s_x (G). Some general properties satisfied by this concept are studied. The x - Steiner numbers of certain classes of graphs are determined. Connected graphs of order p with x - Steiner number 1 or p-1 are characterized. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G such that s(G) = a and s_x (G)= b for some vertex x in G, where s(G) is the Steiner number of a graph.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41601672","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 : 2020-04-18DOI: 10.22108/TOC.2020.122329.1719
A. Lucchini
We prove that the graph obtained from the non-nilpotent graph of a finite group by deleting the isolated vertices is connected with diameter at most 3. This bound is the best possible.
我们证明了从有限群的非幂零图中删除孤立顶点得到的图的直径至多为3。这个界限是最好的。
{"title":"The diameter of the non-nilpotent graph of a finite group","authors":"A. Lucchini","doi":"10.22108/TOC.2020.122329.1719","DOIUrl":"https://doi.org/10.22108/TOC.2020.122329.1719","url":null,"abstract":"We prove that the graph obtained from the non-nilpotent graph of a finite group by deleting the isolated vertices is connected with diameter at most 3. This bound is the best possible.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"111-114"},"PeriodicalIF":0.4,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48656618","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 : 2020-03-06DOI: 10.22108/TOC.2020.120375.1689
Muhammad Aamer Rashid, Sarfraz Ahmad, M. Hanif, M. K. Siddiqui, M. Naeem
A zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum $k$-flow if the absolute values of edges are less than $k$. We define the zero-sum flow number of $G$ as the least integer $k$ for which $G$ admitting a zero sum $k$-flow.? In this paper we gave complete zero-sum flow and zero sum numbers for categorical and strong product of two graphs namely cycle and paths.
{"title":"Zero-Sum Flow Number of Categorical and Strong Product of Graphs","authors":"Muhammad Aamer Rashid, Sarfraz Ahmad, M. Hanif, M. K. Siddiqui, M. Naeem","doi":"10.22108/TOC.2020.120375.1689","DOIUrl":"https://doi.org/10.22108/TOC.2020.120375.1689","url":null,"abstract":"A zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum $k$-flow if the absolute values of edges are less than $k$. We define the zero-sum flow number of $G$ as the least integer $k$ for which $G$ admitting a zero sum $k$-flow.? In this paper we gave complete zero-sum flow and zero sum numbers for categorical and strong product of two graphs namely cycle and paths.","PeriodicalId":43837,"journal":{"name":"Transactions on Combinatorics","volume":"9 1","pages":"181-199"},"PeriodicalIF":0.4,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43859917","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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