Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G so that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G is the smallest t so that there is such a k-tuple coloring of G using t colors in all. The Kneser graph K(m,n) has as vertices all n-element subsets of the set {1,2, . . . ,m} and an edge between two ...

a {em roman dominating function} on a graph $g$ is a function$f:v(g)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}a {em restrained roman dominating}function} $f$ is a {color{blue} roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} the wei...

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V , |NG[v]∩S| ≥ k. Also the total k-domination number γ×k,t(G) of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V , |NG(v)∩S| ≥ k. The k-transversal numb...

Given positive integers n, k, and m, the (n, k)-th mrestrained Stirling number of the first kind is the number of permutations of an n-set with k disjoint cycles of length ≤ m. Inverting the matrix consisting of the (n, k)-th m-restrained Stirling number of the first kind as the (n+1, k +1)-th entry, the (n, k)-th m-restrained Stirling number of the second kind is defined. In this paper, the mu...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...