University of IsfahanTransactions on Combinatorics2251-86579420201201Zero-sum flow number of categorical and strong product of graphs1811992451710.22108/toc.2020.120375.1689ENMuhammad AamerRashidDepartment of Mathematics,
COMSATS University Islamabad, Lahore Campus, 54000, PakistanSarfrazAhmadDepartment of Mathematics, COMSATS University Islamabad, Lahore Campus, 54000, PakistanMuhammad FarhanHanifDepartment of Mathematics,
COMSATS University Islamabad, Lahore Campus, 54000, PakistanMuhammad KamranSiddiquiDepartment of Mathematics COMSATS University Islamabad, Lahore Campus, 54000, PakistanMuhammadNaeemDepartment of Mathematics, The University of Okara, PakistanJournal Article20191207A 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579420201201Further results on maximal rainbow domination number2012102457910.22108/toc.2020.120014.1684ENHosseinAbdollahzadeh AhangarDepartment of Mathematics, Babol Noshirvani University of Technology, Babol, I.R. IranJournal Article20191111A <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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579420201201Edge-group choosability of outerplanar and near-outerplanar graphs2112162480610.22108/toc.2020.116355.1633ENAmirKhamsehDepartment of Mathematics, Kharazmi University, 15719-14911, Tehran, Iran0000-0001-5077-634XJournal Article20190409Let $chi_{gl}(G)$ 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579420201201On the dominated chromatic number of certain graphs2172302486210.22108/toc.2020.119361.1675ENSaeidAlikhaniDepartment of Mathematics, Yazd University, 89195-741, Yazd, IranMohammad RezaPiriDepartment of Mathematics, Yazd University, 89195-741, Yazd, IranJournal Article20190925Let $G$ 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.pdfUniversity of IsfahanTransactions on Combinatorics2251-86579420201201Gutman index, edge-Wiener index and edge-connectivity2312422486810.22108/toc.2020.124104.1749ENJayaMazorodzeDepartment of Mathematics, University of Zimbabwe, P. O. Box MP 167, Mount Pleasant, Harare, ZimbabweSimonMukwembiSchool of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South AfricaTomasVetrikDepartment of Mathematics and Applied Mathematics, University of the Free State, P. O. Box 339, Bloemfontein, 9300,
South AfricaJournal Article20200723We 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