Resumen de Powers of Hamilton cycles in graphs perturbed by a random geometric graph

Alberto Espuny Díaz, Joseph Hyde

• A random geometric graph Gd(n, r) is obtained by placing n vertices uniformly andindependently at random in the hypercube [0, 1]d and joining two vertices by an edge ifthe distance between them is at most r. We study the problem of the containment of k-thpowers of Hamilton cycles in the union of Gd(n, r) with an n-vertex graph Hn with minimumdegree αn. For all values of k, d and α, we provide asymptotically optimal values for rwhich ensure the union contains the k-th power of a Hamilton cycle with high probability.Our result implies asymptotically optimal conditions for the containment of other spanningstructures.