We analyze classical Maker-Breaker games played on the edge set of a randomly generated graph G.We consider the random graph process and analyze, for each of the properties "being spanning k-vertex-connected" , "admitting a perfect matching", and "being Hamiltonian", the first time when Maker starts having a winning strategy for building a graph possessing the target property (the so called hitting time).We prove that typically it happens precisely at the time the random graph process first reaches minimum degree 2k, 2 and 4, respectively, which is clearly optimal.The latter two statements settle conjectures of Stojaković and Szabó.We also consider a general-purpose game, the expander game, which is a main ingredient of our proofs and might be of an independent interest.

Let [Formula: see text] be a sequence of [Formula: see text] integers. For an increasing monotone graph property [Formula: see text] we say that a base graph [Formula: see text] is [Formula: see text]-resilient with respect to [Formula: see text] if for every subgraph [Formula: see text] such that [Formula: see text] for every [Formula: see text] the graph [Formula: see text] possesses [Formula: see text]. This notion naturally extends the idea of the local resilience of graphs recently initiated by Sudakov and Vu. In this paper we study the [Formula: see text]-resilience of a typical graph from [Formula: see text] with respect to the Hamiltonicity property, where we let [Formula: see text] range over all values for which the base graph is expected to be Hamiltonian. Considering this generalized approach to the notion of resilience our main result implies several corollaries which improve on the best known bounds of Hamiltonicity related questions. For one, it implies that for every positive [Formula: see text] and large enough values of [Formula: see text], if [Formula: see text], then with high probability the local resilience of [Formula: see text] with respect to being Hamiltonian is at least [Formula: see text], improving on the previous bound for this range of [Formula: see text]. Another implication is a result on optimal packing of edge-disjoint Hamilton cycles in a random graph. We prove that if [Formula: see text], then with high probability a graph [Formula: see text] sampled from [Formula: see text] contains [Formula: see text] edge-disjoint Hamilton cycles, extending the previous range of [Formula: see text] for which this was known to hold.