Abstract
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 $\geq q$ 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.
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