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

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 β_k(G). In this paper, we show that for every graph G, β_k(G) ≥ ⌈(n+( χ(G) −1) ∑_{v ∈ S(G)}min(|L_v| ,k −1))/ χ(G) ⌉, where χ(G),s(G) and L_v 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.

References

[1]	S.T. Hedetniemi, Hereditary properties of graphs, J. Combin. Theory (B) 14 (1973) 94--99, doi: 10.1016/S0095-8956(73)80009-7 .
[2]	M. Blidia, M. Chellali, O. Favaron and N. Meddah, On k-independence in graphs with emphasis on trees, Discrete Math. 307 (2007) 2209--2216, doi: 10.1016/j.disc.2006.11.007.
[3]	M. Borowiecki and D. Michalak, Generalized independence and domination in graphs, Discrete Math. 191 (1998) 51--56, doi: 10.1016/S0012-365X(98)00092-2.
[4]	R.L. Brooks, On coloring the nodes of a network, Math. Proc. Cambridge Philos. Soc. 37 (1941) 194--197, doi: 10.1017/S030500410002168X.
[5]	Y. Caro and Z. Tuza, Improved lower bounds on k-independence, J. Graph Theory 15 (1991) 99--107, doi: 10.1002/jgt.3190150110.
[6]	O. Favaron, k-domination and k-independence in graphs, Ars Combin. 25 (1988) 159--167.
[7]	O. Favaron, On a conjecture of Fink and Jacobson concerning k-domination and k-dependence, J. Combin. Theory (B) 39 (1985) 101--102, doi: 10.1016/0095-8956(85)90040-1.
[8]	J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Graph Theory with Applications to Algorithms and Computer Science, John Wiley and Sons, New York (1985) 283--300.
[9]	J.F. Fink and M.S. Jacobson, n-domination, n-dependence and forbidden subgraphs, in: Graph Theory with Applications to Algorithms and Computer Science, John Wiley and Sons, New York (1985) 301--311.
[10]	T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).
[11]	T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc., New York, 1998).
[12]	L. Volkmann, A characterization of bipartite graphs with independence number half their order, Australas. J. Combin. 41 (2008) 219--222.

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