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De Bruijn Covering Codes for Rooted Hypergraphs

Joshua N. Cooper·Fan Chung
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Abstract

What is the length of the shortest sequence $S$ of reals so that the set of consecutive $n$-words in $S$ form a covering code for permutations on $\{1,2, >..., n\}$ of radius $R$ ? (The distance between two $n$-words is the number of transpositions needed to have the same order type.) The above problem can be viewed as a special case of finding a De Bruijn covering code for a rooted hypergraph. Each edge of a rooted hypergraph contains a special vertex, called the {\it root} of the edge, and each vertex is the root of a unique edge, called its {\it ball}. A De Bruijn covering code is a subset of the roots such that every vertex is in some edge containing a chosen root. Under some mild conditions, we obtain an upper bound for the shortest length of a De Bruijn covering code of a rooted hypergraph, a bound which is within a factor of $\log n$ of the lower bound.

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Joshua N. Cooper, Fan Chung. 2005-05-25. De Bruijn Covering Codes for Rooted Hypergraphs. https://arxiv.org/abs/math/0505528

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