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In his seminal 1976 paper, P\'osa showed that for all $p\geq C\log n/n$, thebinomial random graph $G(n,p)$ is with high probability Hamiltonian. This leadsto the following natural questions, which have been extensively studied: Howwell is it typically possible to cover all edges of $G(n,p)$ with Hamiltoncycles? How many cycles are necessary? In this paper we show that for $ p\geqC\log n/n$, we can cover $G\sim G(n,p)$ with precisely$\lceil\Delta(G)/2\rceil$ Hamilton cycles. Our result is clearly best possibleboth in terms of the number of required cycles, and the asymptotics of the edgeprobability $p$, since it starts working at the weak threshold needed forHamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, andimproves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and ofFerber, Kronenberg and Long, essentially closing a long line of research onHamiltonian packing and covering problems in random graphs.

Optimal Hamilton covers and linear arboricity for random graphs