Packing Hamilton Cycles Online

It is known that w.h.p. the hitting time ${\tau }_{2\sigma }$ for the random graph process to have minimum degree $2\sigma$ coincides with the hitting time for $\sigma$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $\sigma \ge 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $\sigma$ colors so that at time ${\tau }_{2\sigma }$, each color class is Hamiltonian.