De Bruijn Covering Codes for Rooted Hypergraphs
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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505528
 arXiv:
 arXiv:math/0505528
 Bibcode:
 2005math......5528C
 Keywords:

 Combinatorics;
 Probability;
 MSCclass: 05D40 (Primary) 68R15;
 05B40 (Secondary)
 EPrint:
 10 pages, no figures