Universal Cycles of Restricted Words

TL;DR

This paper extends the theory of universal cycles by establishing new existence results for various combinatorial objects, including functions and paths, enriching the understanding of their structural properties.

Contribution

It introduces new existence results for universal cycles of monotone, augmented onto, Lipschitz functions, and certain lattice paths and random walks.

Findings

Universal cycles exist for monotone functions

Universal cycles are established for augmented onto functions

Results include universal cycles for specific lattice paths and random walks

Abstract

A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian, this baseline result is used as the basis of existence proofs for universal cycles (also known as generalized deBruijn cycles or U-cycles) of several combinatorial objects. We extend the body of known results by presenting new results on the existence of universal cycles of monotone, "augmented onto", and Lipschitz functions in addition to universal cycles of certain types of lattice paths and random walks.