Discussiones Mathematicae Graph Theory 33(2) (2013) 395-410
doi: 10.7151/dmgt.1677

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A characterization of trees for a new lower bound on the k-independence number

Nacéra Meddah and Mostafa Blidia

LAMDA-RO, Department of Mathematics
University of Blida
B.P. 270, Blida, Algeria

Abstract

Let k be a positive integer and G = (V,E) a graph of order n. A subset S of V is a k-independent set of G if the maximum degree of the subgraph induced by the vertices of S is less or equal to k −1. The maximum cardinality of a k-independent set of G is the k-independence number βk(G). In this paper, we show that for every graph G, βk(G) ≥ ⌈(n+( χ(G) −1) ∑v ∈ S(G)min(|Lv| ,k −1))/ χ(G) ⌉, where χ(G),s(G) and Lv are the chromatic number, the number of supports vertices and the number of leaves neighbors of v, in the graph G, respectively. Moreover, we characterize extremal trees attaining these bounds.

Keywords: domination, independence, k-independence

2010 Mathematics Subject Classification: 05C69.

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Received 16 May 2011
Revised 17 May 2012
Accepted 31 May 2012