This paper considers the (n, k)-Bernoulli-Laplace model in the case when there are two urns, the total number of red and white balls is the same, and the number of selections k at each step is on the same asymptotic order as the number of balls n in each urn. Our main focus is on the large-time behavior of the corresponding Markov chain tracking the number of red balls in a given urn. Under reasonable assumptions on the asymptotic behavior of the ratio k/n as n → ∞, cutoff in the total variation distance is established. A cutoff window is also provided. These results, in particular, partially resolve an open problem posed by Eskenazis and Nestoridi in [8].

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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.

In this paper arithmetic progressions on the integers and the integers modulo n are extended to graphs. This allows for the definition of the anti-van der Waerden number of a graph. Much of the focus of this paper is on 3-term arithmetic progressions for which general bounds are obtained based on the radius and diameter of a graph. The general bounds are improved for trees and Cartesian products and exact values are determined for some classes of graphs. Larger k-term arithmetic progressions are considered and a connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established.Keywords anti-van der Waerden number; rainbow; k-term arithmetic progression; Ramsey number.