On the k-independence number of graphs

On the k-independence number in graphs

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.

On the signed (total) k-independence number in graphs

Let G be a graph. A function f : V (G) → {−1, 1} is a signed kindependence function if the sum of its function values over any closed neighborhood is at most k − 1, where k ≥ 2. The signed k-independence number of G is the maximum weight of a signed k-independence function of G. Similarly, the signed total k-independence number of G is the maximum weight of a signed total k-independence functio...

A note on k-extendable graphs and independence number

An even order graph G with a perfect matching is k-extendable if for every matching M of size k in G, there exists a perfect matching in G containing all the edges of M. In this note, we estabUsh a necessary and sufficient condition for a graph to be k-extendable in terms of its independence number. All graphs considered in this note are finite, connected, loopless and have no multiple edges. F...

On the independence number of random graphs

On the minimal number of k cliques in graphs with restricted independence number

We investigate the following problem of Erd1⁄2os: Determine the minimal number of k cliques in a graph of order n with independence number l: We disprove the original conjecture of Erd1⁄2os which was stated in 1962 for all but a …nite number of pairs k; l; and give asymptotic estimates for l = 3; k 4 and l = 4; k 4: 1 Introduction Let tm (G) denote the number of complete subgraphs of order m in...