Packing Directed Hamilton Cycles Online.

Consider a directed analogue of the random graph process on n vertices, where the n(n 1) edges are ordered uniformly at random and revealed one at a time. It is known that with high probability (w.h.p.) the first digraph in this process with both in-degree and out-degree >= q has a q-edge-coloring with a Hamilton cycle in each color. We show that this coloring can be constructed online, where each edge must be irrevocably colored as soon as it appears. In a similar fashion, for the undirected random graph process, we present an online n-edge-coloring algorithm which yields w.h.p. q disjoint rainbow Hamilton cycles in the first graph containing q disjoint Hamilton cycles.