Universal Cycles for Some Combinatorial Objects

• Main Author: Campbell, Andre A
• Format: Others
• Published: Digital Commons @ East Tennessee State University 2013
• Subjects: Universal Cycle; de Bruijn Cycle; Posets; Boolean Lattice; Applied Mathematics; Physical Sciences and Mathematics
• Online Access: https://dc.etsu.edu/etd/1130; https://dc.etsu.edu/cgi/viewcontent.cgi?article=2330&context=etd

A de Bruijn cycle commonly referred to as a universal cycle (u-cycle), is a complete and compact listing of a collection of combinatorial objects. In this paper, we show the power of de Bruijn's original theorem, namely that the cycles bearing his name exist for n-letter words on a k-letter alphabet for all values of k,n, to prove that we can create de Bruijn cycles for multi-sets using natural encodings and M-Lipschitz n-letter words and the assignment of elements of [n]={1,2,...,n} to the sets in any labeled subposet of the Boolean lattice; de Bruijn's theorem corresponds to the case when the subposet in question consists of a single ground element. In this paper, we also show that de Bruijn's cycles exist for words with weight between s and t, where these parameters are suitably restricted.