Rainbow hamilton cycles in randomly colored randomly perturbed dense graphs

Given an n-vertex graph H with minimum degree at least dn for some fixed d > 0, the distribution HG(n, p) over the supergraphs of H is referred to as a (random) perturbation of H. We consider the distribution of edge-colored graphs arising from assigning each edge of the random perturbation HG(n, p) a color, chosen independently and uniformly at random from a set of colors of size r := r(n). We prove that edge-colored graphs which are generated in this manner asymptotically almost surely admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies p := p(n) ≥ C/n for some fixed C > 0 and r = (1 + o(1))n. The number of colors used is clearly asymptotically best possible. In particular, this improves on a recent result of Anastos and Frieze [J. Graph Theory, 92 (2019), pp. 405-414] in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-colored sparse pseudorandom graphs asymptotically almost surely admit an almost spanning rainbow path.