Resumen de An analogue of Chvátal’s Hamiltonicity theorem for randomly perturbed graphs

Alexander Allin, Alberto Espuny Díaz

• We consider Hamilton cycles in randomly perturbed graphs, that is, graphs obtained asthe union of a deterministic graph H and a random graph G(n, p). While most research intorandomly perturbed graphs assumes a minimum degree condition on H, here we considerconditions on its degree sequence. Under the assumption of a degree sequence of H which iscomparable with the classical condition of Chv´atal (dependent on a parameter α analogousto the minimum degree condition in typical results in the area), we prove that there existssome constant C = C(α) such that taking p = C/n suffices to a.a.s. obtain a Hamilton cyclein H ∪G(n, p). Our result is best possible both in terms of the degree sequence condition andthe asymptotic value of p, and extends the known results about Hamiltonicity in randomlyperturbed graphs. We also provide results about pancyclicity under the same conditions.