Transactions on Combinatorics
Volume:9 Issue: 4, Dec 2020

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.

‎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.

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 $chi'_{gl}(G)leq D$.

‎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.
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‎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 $2 le lambda le 7$‎.