The signed (k,k) -domatic number of digraphs

et $D$ be a finite and simple digraph with vertex set $V(D)$, and let $f:V(D)rightarrow{-1,1}$ be a two-valued function. If $kge 1$ is an integer and $sum_{xin N^-[v]}f(x)ge k$ for each $vin V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from which arcs go into $v$, then $f$ is a signed $k$-dominating function on $D$. A set ${f_1,f_2,ldots,f_d}$ of distinct signed $k$-dominating functions on $D$ with the property that $sum_{i=1}^df_i(x)le k$ for each $xin V(D)$, is called a signed $(k,k)$-dominating family (of functions) on $D$. The maximum number of functions in a signed $(k,k)$-dominating family on $D$ is the signed $(k,k)$-domatic number on $D$, denoted by $d_{S}^{k}(D)$. In this paper, we initiate the study of the signed $(k,k)$-domatic number of digraphs, and we present different bounds on $d_{S}^{k}(D)$. Some of our results are extensions of well-known properties of the signed domatic number $d_S(D)=d_{S}^{1}(D)$ of digraphs $D$ as well as the signed $(k,k)$-domatic number $d_S^k(G)$ of graphs $G$

The Signed Roman Domatic Number of a Digraph

Let $D$ be a finite and simple digraph with vertex set $V(D)$.A {\em signed Roman dominating function} on the digraph $D$ isa function $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that$\sum_{u\in N^-[v]}f(u)\ge 1$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ andall inner neighbors of $v$, and every vertex $u\in V(D)$ for which $f(u)=-1$ has an innerneighbor $v$ for which $f(v)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signedRoman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each$v\in V(D)$, is called a {\em signed Roman dominating family} (of functions) on $D$. The maximumnumber of functions in a signed Roman dominating family on $D$ is the {\em signed Roman domaticnumber} of $D$, denoted by $d_{sR}(D)$. In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for $d_{sR}(D)$. In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs

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