Shortened universal cycles for permutations

Abstract

Kitaev, Potapov, and Vajnovszki [On shortening u-cycles and u-words for permutations, Discrete Appl. Math, 2019] described how to shorten universal words for permutations, to length n!+n−1−i(n−1) for any i ∈ [(n − 2)!], by introducing incomparable elements. They conjectured that it is also possible to use incomparable elements to shorten universal cycles for permutations to length n! − i(n − 1) for any i ∈ [(n − 2)!]. In this note we prove their conjecture. The proof is constructive, and, on the way, we also show a new method for constructing universal cycles for permutations.

Anti-Ramsey problems on graphs and hypergraphs

“…In [43], the authors extend the result of Kitaev, Potapov, and Vajnovszki's [45] to universal cycles. Namely, introducing incomparable elements at distance n − 1 can obtain universal cycles for S n of certain lengths.…”