We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers k ⩾ 1 , r ⩾ 0, and ℓ ⩾ ( r + 1 ) r, and for any α > k k + 1 we show that adding O ( n 2 − 2 ∕ ℓ ) random edges to an n‐vertex graph G with minimum degree at least α n yields, with probability close to one, the existence of the ( k ℓ + r )‐th power of a Hamiltonian cycle. In particular, for r = 1 and ℓ = 2 this implies that adding O ( n ) random edges to such a graph G already ensures the ( 2 k + 1 )‐st power of a Hamiltonian cycle (proved independently by Nenadov and Trujić). In this instance and for several other choices of k , ℓ, and r we can show that our result is asymptotically optimal.

Abstract. Let Γ(n, p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p ≤ c/n 2 , then a.a.s. Γ(n, p) is free and if p ≥ Clog n/n 2

Abstract. The random triangular group Γ(n, t) is a group given by a presentation P = S|R , where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of Γ(n, t) is in some respects similar to that of widely studied density random group introduced by Gromov. In particular, it is known that if t ≤ n 3/2−ε for some ε > 0, then with probability 1 − o(1) Γ(n, t) is infinite and hyperbolic, while for t ≥ n 3/2+ε , with probability 1 − o(1) it is trivial. In this note we show that Γ(n, t) collapses provided only that t ≥ Cn 3/2 for some constant C > 0.By S|R we denote a presentation P of a group, where S is the set of generators and R the set of relations. A presentation P is a triangular presentation if the set R of relations consists of distinct cyclically reduced words of length three only (over the alphabet S ∪S −1 which consists of generators and their formal inverses). A triangular group is a group given by a triangular presentation.In the paper we investigate the properties of groups given by presentations P = S|R , where S is a set of n generators and R is a random subset of the set T of all N = 2n(4n 2 − 6n + 3) ∼ 8n 3 distinct cyclically reduced words of length three, that is words of the form abc, where a = b −1 , b = c −1 and c = a −1 . Here we consider two models of a random triangular group. In the random uniform model Γ(n, t), the set of relations R is chosen uniformly at random from the family of all N t subsets of T of size t. In the binomial model Γ(n, p), we select each element from T independently with probability p. We shall study the asymptotic properties of these two models as n → ∞. In particular, we say that for a given function t(n) [or p(n)] the group Γ(n, t(n)) [resp. Γ(n, p(n))] has a property a.a.s. (asymptotically almost surely) if the probability that the random group has this property tends to 1 as n → ∞. It should be mentioned that, as is well known in the theory of random structures, the asymptotic behavior of Γ(n, t) and Γ(n, p) can

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Abstract. The random triangular group Γ(n, p) is the group given by a random group presentation with n generators in which every relator of length three is present independently with probability p. We show that in the evolution of Γ(n, p) the property of collapsing to the trivial group admits a very sharp threshold.