Covering Complete Hypergraphs with Cuts of Minimum Total Size |
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Authors: | Sebastian M Cioab? André Kündgen |
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Institution: | 1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA 2. Department of Mathematics, California State University San Marcos, San Marcos, CA, 92096, USA
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Abstract: | A cut X, V − X] in a hypergraph with vertex-set V is the set of all edges that meet both X and V − X. Let s
r
(n) denote the minimum total size of any cover of the edges of the complete r-uniform hypergraph on n vertices Knr{K_n^r} by cuts. We show that there is a number n
r
such that for every n > n
r
, s
r
(n) is uniquely achieved by a cover with
?\fracn-1r-1?{\lfloor \frac{n-1}{r-1}\rfloor} cuts X
i
, V − X
i
] such that the X
i
are pairwise disjoint sets of size at most r − 1. We show that c1r2r < nr < c2r52r{c_1r2^r < n_r < c_2r^52^r} for some positive absolute constants c
1 and c
2. Using known results for s
2(n) we also determine s
3(n) exactly for all n. |
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Keywords: | |
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