In this paper, we deal with the independence number of iterated line digraphs of a regular digraph G. We give pertinent lower bounds and give an asymptotic estimation of the ratio of the number of vertices of a largest independent set of the nth iterated line digraph of G. © 2005 Elsevier B.V. All rights reserved.

By using the notion of compatibility of subgraphs with a perfect matching developed for digraphs in [1], we show that if, in a balanced bipartite graph G of minimum degree 6, the maximum cardinality ebip of a balanced independent subset satisfies ~bip ~< 26-4, then G is hamiltonian-biconnected, and if Ctbip ~< 26-2, G contains a hamiltonian path. Moreover, we give some properties of balanced bi...

For finite graphs F and G, let Nr(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. I f g and c?? are families of graphs, it is natural to ask then whether or not the quantities NF(G), FE F, are linearly independent when G is restricted to Q. For example, if P = (K1, &} (where K, denotes the complete graph on n vertices) and 9 is the fa...

For a non-negative integer T , we prove that the independence number of a graph G = (V,E) in which every vertex belongs to at most T triangles is at least ∑ u∈V f(d(u), T ) where d(u) denotes the degree of a vertex u ∈ V , f(d, T ) = 1 d+1 for T ≥ ( d 2 ) and f(d, T ) = (1 + (d2− d− 2T )f(d− 1, T ))/(d2 + 1− 2T ) for T < ( d 2 ) . This is a common generalization of the lower bounds for the inde...

We consider the problem of approximating the size of a minimum non-extendible independent set of a graph, also known as the minimum dominating independence number. We strengthen a result of Irving [2] to show that there is no constant > 0 for which this problem can be approximated within a factor of n 10 in polynomial time, unless P = NP. This is the strongest lower bound we are aware of for po...

In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, let f (k, ) be the maximum order of a graph G with independence number (G) , which has no k vertex-disjoint cycles. We prove that f (k, ) = 3k + 2 − 3 if 1 5 or 1 k 2, and f (k, ) 3k + 2 − 3 in general. We also prove the following results: (1) there exists a constant c (depending only on ) such that f...

A partial Steiner (n, r, l)-system is an r-uniform hypergraph on n vertices in which every set of l vertices is contained in at most one edge. A partial Steiner (n, r, l)-system is complete if every set of l vertices is contained in exactly one edge. In a hypergraph H, the independence number α(H) denotes the maximum size of a set of vertices in H containing no edge. In this article we prove th...

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.

There are many functions of the degree sequence of a graph which give lower bounds on the independence number of the graph. In particular, for every graph G, α(G) ≥ R(d(G)), where R is the residue of the degree sequence of G. We consider the precision of this estimate when it is applied to semi-regular degree sequences. We show that the residue nearly always gives the best possible estimate on ...