For any k ∈ N, the k-subdivision of a graph G is a simple graph G 1 k , which is constructed by replacing each edge of G with a path of length k. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the mth power of the n-subdivision of G has been introduced as a fractional power of G, denoted by G m n . In this regard, we investig...

A tree T , in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V (T ). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rxk(G). G...

The concept of split domination number was introduced by Kulli and Janakiram. Fink Jacobson the notion k-domination in graphs. In this paper, we acquaint with k-split k-non for some zero-divisor graphs ϑ-Obrazom

let $d$ be a finite and simple digraph with vertex set $v(d)$‎.‎a signed total roman $k$-dominating function (str$k$df) on‎‎$d$ is a function $f:v(d)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each‎‎$vin v(d)$‎, ‎where $n^{-}(v)$ consists of all vertices of $d$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set ofG if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investi...

A vertex in a graph totally dominates another vertex if they are adjacent. A sequence of vertices in a graph G is called a total dominating sequence if every vertex v in the sequence totally dominates at least one vertex that was not totally dominated by any vertex that precedes v in the sequence, and at the end all vertices of G are totally dominated. While the length of a shortest such sequen...

A subset S of a vertex set of a graphG is a total (k, r)-dominating set if every vertex u ∈ V (G) is within distance k of at least r vertices in S. The minimum cardinality among all total (k, r)-dominating sets ofG is called the total (k, r)domination number of G, denoted by γ (k,r)(G). We previously gave an upper bound on γ t (2,r)(G(n, p)) in random graphs with non-fixed p ∈ (0, 1). In this p...

{sl let $[n]={1,dots, n}$ be colored in $k$ colors. a rainbow ap$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. conlon, jungi'{c} and radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow ap(4) free, when $n$ is even. based on their construction, we show that such a coloring of ...

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating ...