In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Pósa and Korshunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.

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Maker-Breaker games are played on a hypergraph pX, F q, where F Ď 2 X denotes the family of winning sets. Both players alternately claim a predefined amount of edges (called bias) from the board X, and Maker wins the game if she is able to occupy any winning set F P F . These games are well studied when played on the complete graph K n or on a random graph G n,p .In this paper we consider Maker-Breaker games played on randomly perturbed graphs instead.These graphs consist of the union of a deterministic graph G α with minimum degree at least αn and a binomial random graph G n,p . Depending on α and Breaker's bias b we determine the order of the threshold probability for winning the Hamiltonicity game and the k-connectivity game on G α Y G n,p , and we discuss the H-game when b " 1. Furthermore, we give optimal results for the Waiter-Client versions of all mentioned games. OP was supported by the DFG (Grant PA 3513/1-1).1 2 DENNIS CLEMENS, FABIAN HAMANN, YANNICK MOGGE, AND OLAF PARCZYK minimum degree at least n`k´1 2 is k-vertex-connected. Again this bound is seen to be sharp by looking at the following example: take the vertex disjoint union of three cliques of sizes k ´1, t n´k`1 2 u and r n´k`1 2 s, respectively, and add all edges between the first clique and both of the other cliques. For the containment of a fixed graph H a sufficient minimum degree condition can be derived from Turán's Theorem [46] with regularity, or from the following theorem of Erdős and Stone [20]: for any fixed graph H with chromatic number r ą 2, any n-vertex graph with minimum degree `r´2 r´1 `op1q ˘n contains a copy of H provided that n is large. This bound is seen to be sharp by considering a Turán graph on n vertices with r ´1 vertex classes, i.e. an pr ´1q-partite graph with all class sizes differing at most by 1. For bipartite graphs H any linear minimum degree is sufficient.1.2. Random graphs. Let G n,p be the binomial random graph model, i.e. the model of n-vertex random graphs, where each edge is present with probability p independently of all others. For simplicity, we will often write G n,p also for a graph G " G n,p drawn according to this distribution.Given some graph property P, we are interested in a probability function p " ppPq such that the following holds: when p " ωppq, then G n,p satisfies P asymptotically almost surely (a.a.s.), i.e. with probability tending to one as n tends to infinity. When additionally it holds that G n,p a.a.s. does not have property P when p " oppq, we call p a threshold probability for the property P. The existence of such a threshold is known for all monotone graph properties P [12]. If the same conclusion even holds with p ě p1 `εqp and p ď p1 ´εqp for any constant ε ą 0, then we call p a sharp threshold probability for the property P.Posá [43] and Korshunov [36] independently proved that G n,p has a sharp threshold for containing