On the independence number in K1, r+1-free graphs

On the independence number in K1, r+1-free graphs

In this paper we use the degree sequence, order, size and vertex connectivity of a K 1,,+ 1 -free graph or of an almost claw-free graph to obtain several upper bounds on its independence number. We also discuss the sharpness of these results.

Minimum independent generalized t-degree and independence number in K1, r+1-free graphs

The minimum independent generalized t-degree of a graph G = (V,E) is ut = min{ IN(H H is an independent set of t vertices of G}, with N(H) = UxtH N(x). In a KI,~+I -free graph, we give an upper bound on u! in terms of r and the independence number CI of G. This generalizes already known results on u2 in KI,,+I-free graphs and on U, in KI,x-free 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 independence number of random graphs

Triangle-free planar graphs with small independence number

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3−ε) for some fixed ε > 0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-fre...