For an integer k ≥ 1 and a graph G = (V,E), a subset S of V is kindependent if every vertex in S has at most k − 1 neighbors in S. The k-independent number βk(G) is the maximum cardinality of a kindependent set of G. In this work, we study relations between βk(G), βj(G) and the domination number γ(G) in a graph G where 1 ≤ j < k. Also we give some characterizations of extremal graphs.

A function f de1ned on the vertices of a graph G = (V; E); f :V → {−1; 0; 1} is a minus dominating function if the sum of its values over any closed neighborhood is at least one. The weight of a minus dominating function is f(V ) = ∑ v∈V f(v). The minus domination number of a graph G, denoted by −(G), equals the minimum weight of a minus dominating function of G. In this paper, a sharp lower bo...

The following fundamental result for the domination number γ(G) of a graph G was proved by Alon and Spencer, Arnautov, Lovász and Payan: γ(G) ≤ ln(δ + 1) + 1 δ + 1 n, where n is the order and δ is the minimum degree of vertices of G. A similar upper bound for the double domination number was found by Harant and Henning [On double domination in graphs. Discuss. Math. Graph Theory 25 (2005) 29–34...

Here and in what follows we assume S is a finite multiset in R d. This means tha t the points in S have "multiplicity". Strict ly speaking, a multiset is a map S : F --, R d and in our case F is finite. Then corekS = M{convS(E) : E C_ F, IF \ El _< k}. From now on we do not say explicitly tha t the sets in question are multisets. This will make the no ta t ion simpler and will not cause confusi...

A well-known result (Heawood [6], Ringel [11], Ringel and Youngs [10]) states that the maximum chromatic number of a graph embedded in a given surface S coincides with the size of the largest clique that can be embedded in S, and that this number can be expressed as a simple formula in the Eulerian genus of S. We study maximum chromatic number of k edge-disjoint graphs embedded in a surface. We...

In this note, we deal with k-arch graphs, a generalization of trees, which contain k-trees as a subclass. We show that the number of vertex-labelled k-arch graphs with n vertices, for a fixed integer k ≥ 1, is (nk)n−k−1. As far as we know, this is a new integer sequence. We establish this result with a one-to-one correspondence relating k-arch graphs and words whose letters are k-subsets of the...

The main purpose of this paper is to study the distribution properties of the k-power free numbers, and give an interesting asymptotic formula for it.

In the plane Z× Z a cell is a unit square and a polyomino is a finite connected union of cells. Polyominoes are defined up to translations. Since they have been introduced by Golomb [20], polyominoes have become quite popular combinatorial objects and have shown relations with many mathematical problems, such as tilings [6], or games [19] among many others. Two of the most relevant combinatoria...

In this paper the combinatorial problem of determining the number of minimal path sets of a consecutive-k-out-of-n: F system is considered. For the cases where k = 2, 3 the explicit formulae are given and for k &ge; 4 a recursive relation is obtained. Direct computation for determining the number of minimal path sets of a consecutive-k-out-of-n: F system for k &ge; 4 remains a difficult task. ...

let $k$ be a positive integer. a subset $s$ of $v(g)$ in a graph $g$ is a $k$-tuple total dominating set of $g$ if every vertex of $g$ has at least $k$ neighbors in $s$. the $k$-tuple total domination number $gamma _{times k,t}(g)$ of $g$ is the minimum cardinality of a $k$-tuple total dominating set of $g$. if$v(g)=v^{0}={v_{1}^{0},v_{2}^{0},ldots ,v_{n}^{0}}$ and $e(g)=e_{0}$, then for any in...