Universal Cycles of Complementary Classes

TL;DR

This paper extends the concept of universal cycles to complementary classes of combinatorial objects, demonstrating their existence for various complex and constrained classes such as alternating words, non-injective functions, and illegal tournament rankings.

Contribution

It introduces the existence of universal cycles for new classes of objects, including class-alternating words and non-injective functions, expanding the scope of U-cycle theory.

Findings

Existence of U-cycles for class-alternating words

Existence of U-cycles for words with repeated letters

Existence of U-cycles for non-onto functions

Abstract

Universal Cycles, or U-cycles, as originally defined by de Bruijn, are an efficient method to exhibit a large class of combinatorial objects in a compressed fashion, and with no repeats. de Bruijn's theorem states that U-cycles for $n$ letter words on a $k$ letter alphabet exist for all $k$ and $n$. Much has already been proved about Universal Cycles for a variety of other objects. This work is intended to augment the current research in the area by exhibiting U-cycles for {\it complementary classes}. Results will be presented that exhibit the existence of U-cycles for class-alternating words such as alternating vowel-consonant (VCVC) words; words with at least one repeated letter (non-injective functions); words with at least one letter of the alphabet missing (functions that are not onto); words that represent illegal tournament rankings; and words that do not constitute "strong" legal computer passwords. As with previous papers pertaining to U-cycles, connectedness proves to be a nontrivial step.