Signed star k-domatic number of a graph

Let $G$ be a simple graph without isolated vertices with vertex set $V(G)$ and edge set $E(G)$ and let $k$ be a positive integer. A function $f:E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star $k$-dominating function on $G$ if $\sum_{e\in E(v)}f(e)\ge k$ for every vertex $v$ of $G$, where $E(v)=\{uv\in E(G)\mid u\in N(v)\}$. A set $\{f_1,f_2,\ldots,f_d\}$ of signed star $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(e)\le 1$ for each $e\in E(G)$, is called a signed star $k$-dominating family (of functions) on $G$. The maximum number of functions in a signed star $k$-dominating family on $G$ is the signed star $k$-domatic number of $G$, denoted by $d_{kSS}(G)$

Signed total double Roman dominatıon numbers in digraphs

Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) P x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consist of all in-neighbors of v, and (ii) if f(v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x∈V (D) f(x). The signed total double Roman domination number (STDRD-number) γtsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for γtsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.Publisher's Versio

Limited packings: related vertex partitions and duality issues

A $k$-limited packing partition ($k$LP partition) of a graph $G$ is a partition of $V(G)$ into $k$-limited packing sets. We consider the $k$LP partitions with minimum cardinality (with emphasis on $k=2$). The minimum cardinality is called $k$LP partition number of $G$ and denoted by $\chi_{\times k}(G)$. This problem is the dual problem of $k$-tuple domatic partitioning as well as a generalization of the well-studied $2$-distance coloring problem in graphs. We give the exact value of $\chi_{\times2}$ for trees and bound it for general graphs. A section of this paper is devoted to the dual of this problem, where we give a solution to an open problem posed in $1998$. We also revisit the total limited packing number in this paper and prove that the problem of computing this parameter is NP-hard even for some special families of graphs. We give some inequalities concerning this parameter and discuss the difference between $2$TLP number and $2$LP number with emphasis on trees

Total kk-Rainbow domination numbers in graphs

Let $k\geq 1$ be an integer‎, ‎and let $G$ be a graph‎. ‎A {\it‎ ‎$k$-rainbow dominating function} (or a {\it $k$-RDF}) of $G$ is a‎ ‎function $f$ from the vertex set $V(G)$ to the family of all subsets‎ ‎of $\{1,2,\ldots‎ ,‎k\}$ such that for every $v\in V(G)$ with‎ ‎$f(v)=\emptyset$‎, ‎the condition $\bigcup_{u\in‎ ‎N_{G}(v)}f(u)=\{1,2,\ldots,k\}$ is fulfilled‎, ‎where $N_{G}(v)$ is‎ ‎the open neighborhood of $v$‎. ‎The {\it weight} of a $k$-RDF $f$ of‎ ‎$G$ is the value $\omega (f)=\sum _{v\in V(G)}|f(v)|$‎. ‎A $k$-rainbow‎ ‎dominating function $f$ in a graph with no isolated vertex is called‎ ‎a {\em total $k$-rainbow dominating function} if the subgraph of $G$‎ ‎induced by the set $\{v \in V(G) \mid f (v) \not =\emptyset\}$ has no isolated‎ ‎vertices‎. ‎The {\em total $k$-rainbow domination number} of $G$‎, ‎denoted by‎ ‎$\gamma_{trk}(G)$‎, ‎is the minimum weight of a total $k$-rainbow‎ ‎dominating function on $G$‎. ‎The total $1$-rainbow domination is the‎ ‎same as the total domination‎. ‎In this paper we initiate the‎ ‎study of total $k$-rainbow domination number and we investigate its‎ ‎basic properties‎. ‎In particular‎, ‎we present some sharp bounds on the‎ ‎total $k$-rainbow domination number and we determine the total‎ ‎$k$-rainbow domination number of some classes of graphs‎.