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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.

Universal cycles, such as De Bruijn cycles, are cyclic sequences of symbols that represent every combinatorial object from some family exactly once as a consecutive subsequence. Graph universal cycles are a graph analogue of universal cycles introduced in 2010. We introduce graph universal partial cycles, a more compact representation of graph classes, which use "do not know" edges. We show how to construct graph universal partial cycles for labeled graphs, threshold graphs, and permutation graphs. For threshold graphs and permutation graphs, we demonstrate that the graph universal cycles and graph universal partial cycles are closely related to universal cycles and compressed universal cycles, respectively. Using the same connection, for permutation graphs, we define and prove the existence of an s-overlap form of graph universal cycles. We also prove the existence of a generalized form of graph universal cycles for unlabeled graphs.

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