a {em roman dominating function} on a graph $g = (v ,e)$ is a function $f : vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. the {em weight} of a roman dominating function is the value $w(f)=sum_{vin v}f(v)$. the roman domination number of a graph $g$, denoted by $gamma_r(g)$, equals the...

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has at least one neighbor in each of the other color classes. The largest integer k(>0) for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. In this paper, we obtain the b-chromatic number of the square...

Given a graph G, b coloring seeks proper of G with the descriptive property that every color class in contains dominating vertex. We have to minimize number colors coloring. In this paper, we show how chromatic and b-chromatic changes operators relation between has been explored.

Let f be a function that assigns to each vertex a subset of colors chosen from a set C = {1, 2, . . . , k} of k colors. If  u∈N(v) f (u) = C for each vertex v ∈ V with f (v) = ∅, then f is called a k-rainbow dominating function (kRDF) of G where N(v) = {u ∈ V | uv ∈ E}. The weight of f , denoted by w(f ), is defined as w(f ) =  v∈V |f (v)|. Given a graph G, the minimum weight among all weight...

The Square Coloring of a graph G refers to coloring vertices such that any two distinct which are at distance most receive different colors. In this paper, we initiate the study related problem called subset square graphs. Broadly, studies dominating set using q Here aim is optimize number colors used. This also generalizes well-studied Efficient Dominating Set problem. We show $$q$$ -Subset NP...

We first consider some problems related to the maximum number of dominating (or strong dominating) sets in a regular graph. Our techniques, centered around Shearer’s entropy lemma, extend to a reasonably broad class of graph parameters enumerating vertex colorings that satisfy conditions on the multiset of colors appearing in neighborhoods (either open or closed). Dominating sets and strong dom...

Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of n players is assigned a hat from a given color set. Simultaneously, each player tries to guess the color of his/her own hat by looking at colors of hats worn by other players. In this paper, we consider a new variati...