On Universal Cycles for new Classes of Combinatorial Structures

TL;DR

This paper proves the existence of universal cycles for various combinatorial structures, including subsets, matroids, and lattice paths, using natural encodings, and constructs efficient coverings and encodings for these objects.

Contribution

It introduces new methods to establish universal cycles for multiple combinatorial classes and provides explicit constructions and bounds for these cycles.

Findings

Universal cycles exist for subsets, matroids, and lattice paths.

Constructed near-optimal coverings for subsets of fixed size.

Established u-cycles for words over alphabets containing specific characters.

Abstract

A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, matroids, restricted multisets, chains of subsets, multichains, and lattice paths. For subsets, we show that a u-cycle exists for the $k$-subsets of an $n$-set if we let $k$ vary in a non zero length interval. We use this result to construct a "covering" of length $(1+o(1))$$n \choose k$ for all subsets of $[n]$ of size exactly $k$ with a specific formula for the $o(1)$ term. We also show that u-cycles exist for all $n$-length words over some alphabet $\Sigma,$ which contain all characters from $R \subset \Sigma.$ Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets.