The square of a Hamilton cycle in randomly perturbed graphs

Abstract

We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given α ∈ (0, 1), the union of any n-vertex graph with minimum degree αn and the binomial random graph G(n, p). This is known when α > 1 2, and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each α ≤ 1 2. We demonstrate that, as α ranges over the interval (0, 1), the threshold performs a countably infinite number of 'jumps'. Our result has implications on the perturbed threshold for 2-universality, where we also fully address all open cases.

Powers of Hamiltonian cycles in randomly augmented Dirac graphs—The complete collection

“…(This result was independently obtained by Nenadov and Trujić in [8]). For m = 2, Böttcher, Parczyk, Sgueglia, and Skokan in [3] extended this result to a more general setting when δ G αn ( ) ≥ , for all values of α (0, 1) ∈ . However, a vast majority of pairs k m ( , ) were still unaccounted for.…”

“…Let us now define this crucial graph. It has been first used in [2] to prove a lower bound on d 0 , and then in many other papers in the same context, for example, [1,3,5,8]. Definition 2.3.…”

Hamiltonicity of graphs perturbed by a random regular graph

“…Other properties that have been studied in the context of randomly perturbed graphs are, e.g., the existence of powers of Hamilton cycles [2,8,11,17,28], F-factors [3,9,10,20], spanning bounded degree trees [7,25] and (almost) unbounded degree trees [22], or general bounded degree spanning graphs [8]. The model of randomly perturbed graphs has also been extended to other settings.…”