Caro, Y. and Tuza, Z., 1991, Improved lower bounds on k-independence, Journal of Graph Theory, 15, 99--107.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is also of much interest to find ISs that have a guaranteed size. What is known in this context Recall that given a hypergraph H = V; E) the degree of a vertex is the number of edges that it lies in; also, H is called k uniform if all the edges have exactly k elements. Caro and Tuza showed in [6] that for any k 2, any k uniform hypergraph H contains an IS of size at least ff k (H) X v2V 1 d(v) 1= k Gamma 1) d(v) 1) here and from now on, d(v) denotes the degree of v 2 V . For any integer l 0 and real r, Gamma r l Delta is defined to be r(r Gamma 1) Delta ....

Y. Caro and Z. Tuza. Improved lower bounds on k-independence. J. Graph Theory, 15, pp. 99--107, 1991.

....all edges in E have size k. A set I V is called independent if 2 I E = i.e. the set I contains no edge of E. The maximal size of an independent set of H is defined as the independence number ff(H) Caro and Tuza proved the following result, which is an extension of Theorem 1. Theorem 2 [CT] Let H = V; E) be a k uniform hypergraph with k 2. Then ff(H) X v2V f(d(v) where d(v) is the degree of v, i.e. the number of edges containing v and the function f is given by f(d) d Y i=1 1 Gamma 1 i (k Gamma 1) 1 : In fact the result of Caro and Tuza is slightly more ....

....deleted vertices. Thus F (H) F (I) jIj ff(H) 2 We remark that the proof implies a polynomial algorithm that computes an independent set of size at least F (H) in an arbitrary hypergraph H of constant rank. In particular, for uniform hypergraph, this is the so called max algorithm (see also [CT,G]) Successively remove vertices of maximum degree with all incident edges until no edges are left. It is easy to see that a vertex x with maximum degree in a uniform hypergraph has always the property F (H n x) F (H) 3 Proofs of Lemmas For the proof of Lemma 5 we need Lemma 6 Let r 2 N, d 2 ....

Y. Caro, Z. Tuza, Improved Lower Bounds on k-Independence, J. of Graph Theory, (1991), Vol. 15, p. 99-107.

....edges in E have size k. A set I V is called independent if 2 I E = i.e. the set I contains no edge of E. The maximal size of an independent set of H is defined as the independence number ff(H) Caro and Tuza proved the following result, which is an extension of Theorem 1. Theorem 2. [2] Let H = V; E) be a k uniform hypergraph with k 2. Then ff(H) X v2V f(d(v) where d(v) is the degree of v, i.e. the number of edges containing v, and the function f is given by f(d) d Y i=1 1 Gamma 1 i (k Gamma 1) 1 : Remark. The function f in Theorem 2 can be ....

....the choice of the deleted vertices. Thus F (H) F (I) jI j ff(H) We remark that the proof implies a polynomial algorithm that computes an independent set of size at least F (H) in an arbitrary hypergraph H of constant rank. This is a natural extension of the so called MAX algorithm (see also [2, 3]) for uniform hypergraphs: successively remove vertices of maximum degree with all incident edges until no edges are left. It is easy to see that a vertex x with maximum degree in a uniform hypergraph has always the property F (H n x) F (H) Lemma 1. Let r 2 N, C 1 ; C 2 ; C r 0 and C 0 ....

Y. Caro, Z. Tuza, Improved Lower Bounds on k-Independence, J. of Graph Theory, (1991), Vol. 15, p. 99-107.

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Caro, Y. and Tuza, Z., 1991, Improved lower bounds on k-independence, Journal of Graph Theory, 15, 99--107.

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