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By Noga (il-tlav-dm) Bollobás, Béla (-memp) Kim, Jeong Han (-msft and Van H. (-ucsd


Economical covers with geometric applications. (English summary) Proc. London Math. Soc. (3) 86 (2003), no. 2, 273–301. Let H = (V, E) be a hypergraph. H is called k-uniform if every edge consists of k vertices. H is called d-regular if the degree of any vertex is d. Finally, for u, v ∈ V the codegree of u and v is the number of edges containing u and v. A cover of H is a set of its edges containing all of its vertices. A result of Vu states that any k-uniform d-regular hypergraph on n vertices with maximum codegree c has a cover of size n/k + O(n(c/d) 1 k−1 log 3 2 d). In Vu’s result it is assumed that k is a constant. The main result of this paper is an analog of the above bound for the case of k growing with d. It is shown, for instance, that if k> 4 and satisfies the condition e2kc = o(d / log d), then there exists a cover of size n 1

Year: 2010
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