University of Isfahan Transactions on Combinatorics 2251-8657 9 4 2020 12 01 Zero-sum flow number of categorical and strong product of graphs 181 199 24517 10.22108/toc.2020.120375.1689 EN Muhammad Aamer Rashid Department of Mathematics, COMSATS University Islamabad, Lahore Campus, 54000, Pakistan Sarfraz Ahmad Department of Mathematics, COMSATS University Islamabad, Lahore Campus, 54000, Pakistan Muhammad Farhan Hanif Department of Mathematics, COMSATS University Islamabad, Lahore Campus, 54000, Pakistan Muhammad Kamran Siddiqui Department of Mathematics COMSATS University Islamabad, Lahore Campus, 54000, Pakistan Muhammad Naeem Department of Mathematics, The University of Okara, Pakistan Journal Article 2019 12 07 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.?<br />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. https://toc.ui.ac.ir/article_24517_97a6195eb2729b9bbe843c44360c7b64.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 4 2020 12 01 Further results on maximal rainbow domination number 201 210 24579 10.22108/toc.2020.120014.1684 EN Hossein Abdollahzadeh Ahangar Department of Mathematics, Babol Noshirvani University of Technology, Babol, I.R. Iran Journal Article 2019 11 11 ‎A  <em>2-rainbow dominating function</em> (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 ‎ ‎<em>maximal 2-rainbow dominating function</em> 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<em> ‎weight</em> of a maximal 2RDF \$f\$ is the value \$omega(f)=sum_{vin‎ ‎V}|f (v)|\$‎. ‎The  <em>maximal \$2\$-rainbow domination number</em> 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. https://toc.ui.ac.ir/article_24579_674b7d663d3699b5f6163ab85e4b0a02.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 4 2020 12 01 Edge-group choosability of outerplanar and near-outerplanar graphs 211 216 24806 10.22108/toc.2020.116355.1633 EN Amir Khamseh Department of Mathematics, Kharazmi University, 15719-14911, Tehran, Iran 0000-0001-5077-634X Journal Article 2019 04 09 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\$. https://toc.ui.ac.ir/article_24806_069756990b05736c896cb48109b43257.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 4 2020 12 01 On the dominated chromatic number of certain graphs 217 230 24862 10.22108/toc.2020.119361.1675 EN Saeid Alikhani Department of Mathematics, Yazd University, 89195-741, Yazd, Iran Mohammad Reza Piri Department of Mathematics, Yazd University, 89195-741, Yazd, Iran Journal Article 2019 09 25 ‎Let \$G\$ be a simple graph‎. ‎The dominated coloring of \$G\$ is a proper coloring of \$G\$ such that each color class is dominated by at least one vertex‎. ‎The minimum number of colors needed for a dominated coloring of \$G\$ is called the dominated chromatic number of \$G\$‎, ‎denoted by \$chi_{dom}(G)\$‎. ‎Stability (bondage number) of dominated chromatic number of \$G\$ is the minimum number of vertices (edges) of \$G\$ whose removal changes the dominated chromatic number of \$G\$‎. ‎In this paper‎, ‎we study the dominated chromatic number‎, ‎dominated stability and dominated bondage number of certain graphs‎.<br />‎ https://toc.ui.ac.ir/article_24862_7ee6f29ca5ca8795e3a390c8fe8145c4.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 4 2020 12 01 Gutman index‎, ‎edge-Wiener index and edge-connectivity 231 242 24868 10.22108/toc.2020.124104.1749 EN Jaya Mazorodze Department of Mathematics, University of Zimbabwe, P. O. Box MP 167, Mount Pleasant, Harare, Zimbabwe Simon Mukwembi School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa Tomas Vetrik Department of Mathematics and Applied Mathematics, University of the Free State, P. O. Box 339, Bloemfontein, 9300, South Africa Journal Article 2020 07 23 ‎We study the Gutman index \${rm Gut}(G)\$ and the edge-Wiener index \$W_e (G)\$ of connected graphs \$G\$ of given order \$n\$ and edge-connectivity \$lambda\$‎. ‎We show that the bound \${rm Gut}(G) le frac{2^4 cdot 3}{5^5 (lambda+1)} n^5‎ + ‎O(n^4)\$ is asymptotically tight for \$lambda ge 8\$‎. ‎We improve this result considerably for \$lambda le 7\$ by presenting asymptotically tight upper bounds on \${rm Gut}(G)\$ and \$W_e (G)\$ for \$2 le lambda le 7\$‎. https://toc.ui.ac.ir/article_24868_8bc0894fb3e9c5068871d370376a8469.pdf