An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow kconnectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that any two distinct vertices of G are connected by k in...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

a set $s$ of vertices in a graph $g$ is a dominating set if every vertex of $v-s$ is adjacent to some vertex in $s$. the domination number $gamma(g)$ is the minimum cardinality of a dominating set in $g$. the annihilation number $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this p...

Abstract. Let G be a nontrivial connected graph on which is defined a coloring N k k G E c ∈ → }, ,...., 3 , 2 , 1 { ) ( : , of the edges of G, where adjacent edges may be colored the same. A path in G is called a rainbow path if no two edges of it are colored the same. G is rainbow connected if G contains a rainbow v u − path for every two vertices u and v in it. The minimum k for which there ...

We propose a memory efficient self-stabilizing protocol building distance-k independent dominating sets. A distance-k independent dominating set is a distance-k independent set and a distance-k dominating set. Our algorithm, named SID, is silent; it converges under the unfair distributed scheduler (the weakest scheduling assumption). The protocol SID is memory efficient : it requires only log(2...

A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson’s Theorem. For every integer k there exists an n0 = n0(k) so that for all n > n0, if n mod k(k − 1) ∈ {1, k}, then every properly edge-colored Kn contains (n 2 ) / (k 2 ) pairwise edge-disjoint rainbow copies of Kk. Our proof uses, as a main ingredient, a double application ...

Given a graph $G=(V,E)$ and a vertex $v in V$, by $N(v)$ we represent the open neighbourhood of $v$. Let $f:Vrightarrow {0,1,2}$ be a function on $G$. The weight of $f$ is $omega(f)=sum_{vin V}f(v)$ and let $V_i={vin V colon f(v)=i}$, for $i=0,1,2$. The function $f$ is said to bebegin{itemize}item a Roman ${2}$-dominating function, if for every vertex $vin V_0$, $sum_{uin N(v)}f(u)geq 2$. The R...