COLORFUL HAMILTON CYCLES IN RANDOM GRAPHS

Given an n vertex graph whose edges have colored from one of r colors C = {c1, c2, . . . , cr}, we define the Hamilton cycle color profile hcp(G) to be the set of vectors (m1, m2, ... , mr) \in [0, n]r such that there exists a Hamilton cycle that is the concatenation of r paths P1, P2, . . . , Pr, where Pi contains mi edges of color ci. We study hcp(Gn,p) when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when hcp(Gn,p) = {(m1, m2, . . . , mr) \in [0, n]r : m1 + m2 + \cdot \cdot \cdot + mr = n}.