Resilient degree sequences with respect to Hamilton cycles and matchings in random graphs

P\'osa's theorem states that any graph $G$ whose degree sequence $d_1 \le \ldots \le d_n$ satisfies $d_i \ge i+1$ for all $i<n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e. we prove a `resilience version' of P\'osa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chv\'atal's theorem generalises P\'osa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chv\'atal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.


Introduction
One of the most well-known and well-studied properties in graph theory is Hamiltonicity. We say that a graph G is Hamiltonian whenever it contains a cycle which covers all of the vertices of G. We refer to such a cycle as a Hamilton cycle. The problem of determining whether or not a graph is Hamiltonian is NP-complete [19]. Thus, the study of Hamiltonicity focuses on finding sufficient conditions, particularly in the form of degree conditions.
In 1952, Dirac [11] proved that every n-vertex graph G with minimum degree at least n/2 is Hamiltonian. Pósa [29] strengthened this result. More specifically, a graph G with degree sequence d 1 ≤ . . . ≤ d n such that d i ≥ i + 1 for all i < n/2 is Hamiltonian. This is best possible in the sense that the condition d i ≥ i + 1 cannot be reduced for any i. Chvátal [9] generalised this further by essentially characterising all degree sequences which guarantee Hamiltonicity: a graph with degree sequence d 1 ≤ . . . ≤ d n is Hamiltonian if for all i < n/2 we have d i ≥ i + 1 or d n−i ≥ n − i.
The search for Hamilton cycles in random graphs has also been at the core of the subject (as well as the closely related problem of finding perfect matchings). Erdős and Rényi [12,13] showed that the random graph G n,p with p ≥ C log n/n a.a.s. contains a perfect matching (if n is even and C is large enough). Pósa [30] and Koršunov [22] independently showed that for the same threshold G n,p is a.a.s. Hamiltonian, and Komlós and Szemerédi [21] determined the exact threshold for p. Remarkably, one can strengthen these results to obtain the following hitting time results. Consider the following random graph process: given a vertex set of size n, add each of the n 2 possible edges, one by one, chosen uniformly at random among all edges that have not been added yet. Then, Bollobás and Thomason [8] showed that a.a.s. a perfect matching appears as soon as every vertex has degree at least 1, and Ajtai, Komlós and Szemerédi [1] and Bollobás [7] independently proved that a.a.s. a Hamilton cycle appears as soon as this graph has minimum degree 2.
One more recent approach to extend the classical extremal results to random graphs is based on the following concept of resilience. The local resilience of a graph G with respect to some property P is the maximum number r such that for any subgraph H ⊆ G with ∆(H) < r, the graph G \ H satisfies P. One may view this concept as a measure of the damage an adversary can commit at each vertex of G, without destroying the property P. The systematic study of local resilience was initiated by Sudakov and Vu [33]. Restated in this terminology, Dirac's theorem says that the local resilience of the complete graph K n with respect to Hamiltonicity is ⌊n/2⌋. This concept of resilience naturally suggests a generalisation of Dirac's theorem in the setting of random graphs. Lee and Sudakov [25] proved that, when p = C log n/n and C is sufficiently large, the local resilience of the random graph G n,p with respect to Hamiltonicity is a.a.s. at least (1/2 − ε)np, extending Dirac's theorem to random graphs. This improved on earlier bounds [6,15,33]. Very recently, Montgomery [27] as well as Nenadov, Steger and Trujić [28] independently obtained a hitting time version of this result (Nenadov, Steger and Trujić also obtained such a hitting time version for perfect matchings [28]).
Resilience of random graphs with respect to other properties has also been extensively studied. In particular, the containment of cycles of all possible lengths [23], k-th powers of cycles of all possible lengths [31], bounded degree trees [3], triangle factors [4], and bounded degree graphs [2,17] have been considered. Local resilience with respect to Hamiltonicity has also been studied in other random graph models, such as binomial random directed graphs [14,16,26] and random regular graphs [5,10].
Lee and Sudakov [25] asked for a characterisation of the degree sequences for which the random graph G n,p is resilient with respect to Hamiltonicity, for p close to log n/n. In this paper, we partially answer this question by extending Pósa's theorem to the setting of random graphs. We also prove that the obvious extension to a Chvátal-type degree condition is false, while some modifications to those conditions suffice to force at least the containment of a perfect matching. We conjecture that such a modification is also sufficient for Hamiltonicity.
To state our results precisely, we start with the following definition, which generalises the class of graphs whose degree sequences satisfy Pósa's condition to the setting of random graphs. Definition 1.1 (Pósa-resilience). Let G = G n,p and ε > 0. Let H ε n,p be the collection of all n-vertex graphs H which satisfy the following property: there is an ordering v 1 , . . . , v n of the (1.1) We denote H ε n,p (G) := {H ∈ H ε n,p : H ⊆ G}. We say that G is ε-Pósa-resilient with respect to a property P if G \ H ∈ P for all H ∈ H ε n,p (G). We can now state our first main result. Theorem 1.2. For every ε > 0, there exists C > 0 such that, for p ≥ C log n/n, a.a.s. the random graph G n,p is ε-Pósa-resilient with respect to Hamiltonicity.
Next, we consider the following definition, which generalises the class of graphs whose degree sequences satisfy Chvátal's condition to the setting of random graphs. Definition 1.3 (Chvátal-resilience). Let G = G n,p and ε > 0. Let H ε,0 n,p be the collection of all n-vertex graphs H which satisfy the following property: there is an ordering v 1 , . . . , v n of the vertices with d H (v 1 ) ≥ . . . ≥ d H (v n ) such that, for all i < n/2, either We denote H ε,0 n,p (G) := {H ∈ H ε,0 n,p : H ⊆ G}. We say that G is ε-Chvátal-resilient with respect to a property P if G \ H ∈ P for all H ∈ H ε,0 n,p (G).
Surprisingly, unlike the case of Pósa-resilience, random graphs are not Chvátal-resilient with respect to even the containment of perfect matchings. (We actually prove a stronger result, see Theorem 3.1.) Theorem 1.4. For every 0 < ε < 10 −6 there exists C > 0 such that, for C log n/n ≤ p ≤ 1/25, a.a.s. the random graph G n,p is not ε-Chvátal-resilient with respect to containing a perfect matching.
This leads to the following modified version of Definition 1.3. A related concept (i.e. a shift in the Chvátal condition) was considered by Kühn, Osthus and Treglown [24] in the setting of directed Hamilton cycles. Definition 1.5 (Shifted Chvátal-resilience). Let G = G n,p and let ε, δ > 0. Let H ε,δ n,p be the collection of all n-vertex graphs H which satisfy the following property: there is an ordering v 1 , . . . , v n of the vertices with We denote H ε,δ n,p (G) := {H ∈ H ε,δ n,p : H ⊆ G}. We say that G is (ε, δ)-Chvátal-resilient with respect to a property P if G \ H ∈ P for all H ∈ H ε,δ n,p (G). Note that (1.3) is never satisfied for i < εn. The conditions (1.2) and (1.
for all H ∈ H ε,δ n,p and all vertices v of H. As H ε n,p ⊆ H ε,δ n,p , the same bound holds when considering ε-Pósa-resilience.
With this new definition of shifted Chvátal-resilience we can obtain the following version of Chvátal's theorem for random graphs with respect to the containment of perfect matchings. Theorem 1.6. For every ε > 0, there exists C > 0 such that, for p ≥ C log n/n, a.a.s. the random graph G n,p is (ε, ε)-Chvátal-resilient with respect to containing a perfect matching if n is even.
We conjecture that Theorem 1.6 also holds if perfect matchings are replaced by Hamilton cycles. Conjecture 1.7. For every ε > 0, there exists C > 0 such that, for p ≥ C log n/n, a.a.s. the random graph G n,p is (ε, ε)-Chvátal-resilient with respect to Hamiltonicity.
The following simple construction shows that this statement, if true, is essentially best possible. Let G = G n,p with p ≥ C log n/n for some sufficiently large C. Given any εn ≤ i < n/2, fix disjoint sets X, Y ⊆ V of sizes i and n − i, respectively, and let H be the induced bipartite subgraph between X and Y . One can then prove that a.a.s.
for all x ∈ X and y ∈ Y . Thus, H is 'close' to satisfying the conditions of Definition 1.5, and it is clear that G \ H is not Hamiltonian since it is disconnected. The same construction shows that Theorem 1.2 is essentially best possible (in the sense that we cannot significantly relax the degree condition) and that Theorem 1.6 is essentially best possible when considering odd i. Investigating resilience with respect to degree sequences is natural not only for perfect matchings and Hamilton cycles, but also for other properties. Several results on degree sequences forcing given substructures have been obtained in the classical setting (see e.g. [32,34] for such results involving Pósa-type degree sequences and [20] for Chvátal-type degree sequences). It would be interesting to see if one can obtain resilience versions (for random graphs) of some of these results.
We use a.a.s. as an abbreviation for asymptotically almost surely. Whenever we claim that a result holds a.a.s. for G n,p , we mean that the probability that our result holds tends to one as n tends to infinity. For the purpose of clarity, we will ignore rounding issues when dealing with asymptotic statements, whenever the values we consider tend to infinity with n.
Given an n-vertex graph We denote the minimum degree in a set of vertices as δ G (A) := min{d G (v) : v ∈ A}, and the maximum degree as ∆(G) := max{d G (v) : v ∈ V (G)}. We often consider the sequence of degrees of the vertices of G ordered increasingly, and refer to it as the degree sequence of G.
The binomial random graph G n,p is obtained by adding each of the edges of a complete graph on n vertices with probability p, independently of the other edges. We will always denote the vertex set of G n,p by V . We use G n,m,p for a random bipartite graph with vertex classes of size n and m, respectively; each edge between the classes is added with probability p independently of every other edge, as above. Whenever we consider a random bipartite graph between vertex sets A and B, we also refer to this model as G A,B,p .
Lemma 2.1. Let X be the sum of n independent Bernoulli random variables and let µ : The following lemmas are standard results for random graphs. They can be proved using Chernoff bounds and the fact that the considered random variables follow binomial distributions.

Lemma 2.2.
There exist constants C, c > 0 such that for any p ≥ C log n/n the random graph For every η > 0, there exists a constant C such that, for p ≥ C log n/n, the random graph We now prove some properties of the subgraphs of the random graphs which satisfy the conditions of Definition 1.5. Proposition 2.5. For every 0 < ε < 1, there exists C > 0 such that for p ≥ C log n/n the random graph G = G n,p a.a.s. satisfies that, for all H ∈ H ε,ε n,p (G) and G ′ := G \ H, the following hold: In particular, and for all X, Y ⊆ V with |X| ≥ n(log n) −1/2 and |Y | ≥ ηn we have Throughout the proof, we condition on the event that (2.1) (iii). Condition on the event that statements (i) and (ii) hold, in addition to (2.1) and (2.2). Assume that G ′ is not connected, and let X ⊆ V be a (connected) component of G ′ such that |X| ≤ n/2. Note that |N G ′ (X)| = |X|. As (i) and (ii) both hold, it is easy to see that |X| ≥ εn/2. Let m := |X| − εn/4 ≥ εn/4.
As H ∈ H ε,ε n,p , by Definition 1.5 there exists a labelling v 1 , .

Chvátal-type resilience for matchings in random graphs
Proof of Theorem 1.6. Let 0 < 1/n ≪ 1/C ≪ η ≪ ε ≪ 1 and 1/c < 1, where n is even and c is the constant given by Lemma 2.2. We condition on the event that G = G n,p satisfies the assertions of Lemma 2.2, Lemma 2.3 and Proposition 2.5 with the chosen constants ε, η, C and c, which happens a.a.s. We will show that all such G are (ε, ε)-Chvátal-resilient with respect to containing a perfect matching. Let H ∈ H ε,ε n,p (G) and let G By Tutte's theorem, it suffices to show that, for any vertex set U ⊆ V , the number of odd components of G ′ − U is at most |U | (here a component is odd if it contains an odd number of vertices). As we conditioned on the assertion of Proposition 2.5(iii) and since n is even, this holds if U is the empty set.
Hence, we will prove that, for any non-empty U ⊆ V , the number of (not necessarily odd) components of G ′ − U is at most |U |. As each component of G ′ − U has at least one vertex, we may further assume that |U | < n/2.
Let U ⊆ V with |U | < n/2 and let k be the total number of components of G ′ − U . To derive a contradiction, assume that k > |U |; in particular, k ≥ 2. Enumerate the components in G ′ − U as C 1 , . . . , C k with |C 1 | ≤ |C 2 | ≤ . . . ≤ |C k |. For each S ⊆ [k], let C S := i∈S C i . We consider the cases where |U | is small and large separately.
First, we prove that |C k | is large in this case.
Proof. Suppose otherwise that |C k | ≤ n/2. Let Let S * ∈ S be a set in S with the minimum |C S * |. We claim that |C S * | ≤ n/2. Indeed, suppose this is not the case. Then, we have |S * | ≥ 2. As a partition of S * into two non-empty sets yields two disjoint sets not in S, we have |C S * ∩ D(H)| < 2εn.
It follows that at least one vertex v ∈ D ⊆ C S * is adjacent to a vertex u ∈ C [k]\S * , a contradiction. This proves the claim.
As G ′ satisfies the assertion of Proposition 2.5(ii), we have So suppose that there is an index j ∈ [ℓ] \ [ℓ − εn/8] such that H does not satisfy (1.2) for j. We have that the set C ′′ := C k \ {v 1 , . . . , v n−j−εn } satisfies Here, we obtain the final inequality as |U | ≤ εn/10 and j ≥ ℓ − εn/8. Moreover, because G satisfies the assertion of Lemma 2.3, the fact that (1.3) holds for j implies that As G ′ satisfies the assertion of Proposition 2.5(ii), this shows that a contradiction to the fact that C k is a component of G ′ − U . This proves the claim.
It follows from the previous two claims that G ′ − U has one 'giant' component C k , containing more than (1 − ε/3)n vertices. The following claim will give us the desired contradiction.

This is a contradiction as
We now show that Theorem 1.6 is best possible in the sense that (ε, ε)-Chvátal-resilience cannot be improved to allow for (ε, (3np) −1 )-Chvátal-resilience. That is, unlike the classical theorem of Chvátal, the random graphs analogue requires an extra shift in the indices whenever we veer from a Pósa degree sequence.
Given an n-vertex graph G, we say that G contains an optimal matching if it has a matching of size ⌊n/2⌋. In particular, if G does not contain an optimal matching, then G cannot be Hamiltonian. Note that Theorem 3.1 implies Theorem 1.4. Theorem 3.1. For every 0 < ε < 10 −6 there exists C > 0 such that, for any C log n/n ≤ p ≤ 1/25, the random graph G = G n,p is a.a.s. not (ε, ⌈(3p) −1 ⌉/n)-Chvátal-resilient with respect to containing an optimal matching.
The proof strategy is as follows. We consider G n,p and remove appropriate edges to create a graph G ′ having an independent set X with |N G ′ (X)| < |X| − 1. This ensures that G ′ does not contain an optimal matching. We conclude the proof by showing that G \ G ′ ∈ H ε,⌈(3p) −1 ⌉/n n,p .
(3.5) Note that Lemma 2.3 implies that E 1 happens a.a.s. We condition on the event E 1 . Thus we have Fix disjoint sets X, U ⊆ V \ (Y ∪ N G (Y )) with |X| = 100εn and |U | = |X| − 2. Now, expose all remaining edges of G (i.e. those not incident to Y ). Let E 2 be the event that the following hold for all v ∈ V \ Y and Z ∈ {X, U }: By Lemmas 2.3 and 2.4, the event E 2 happens a.a.s. under conditioning on E 1 . We condition on the event that both E 1 and E 2 hold, i.e. that G satisfies (3.5)-(3.7). We will show that every such G is not (ε, ⌈(3p) −1 ⌉/n)-Chvátal-resilient with respect to containing an optimal matching. We construct a spanning subgraph G ′ of G by deleting all edges in G[X] and all edges in G[X, V \ (X ∪ U )]. From the construction, X is an independent set of G ′ and N G ′ (X) ⊆ U . Thus, |N G ′ (X)| ≤ |U | < |X| − 1, hence G ′ does not contain an optimal matching.
Let γ := ⌈(3p) −1 ⌉/n. Now it suffices to show that H := G \ G ′ ∈ H ε,γ n,p . From the construction, it is easy to see that (3.6) and (3.7) imply that, for all u ∈ Y ∪ U , x ∈ X and v ∈ V \(X ∪ U ∪ Y ), . Hence, H ∈ H ε,γ n,p . Therefore, G n,p a.a.s. contains a subgraph H ∈ H ε,γ n,p such that G n,p \ H does not contain an optimal matching.

Pósa's theorem for Hamilton cycles in random graphs
Our approach for the proof of Theorem 1.2 builds on the ideas of Lee and Sudakov [25], with some modifications and additional steps to account for the increased flexibility in the choice of the graph H that we remove. Thus we only describe the necessary tools as well as the main steps. The corresponding proofs that we omit here can be found in the appendix. For H ∈ H ε n,p , we rely heavily on the fact that graphs of the form G n,p \ H have good expansion properties; namely, they satisfy Proposition 2.5.
Whenever we consider a path P on a vertex set W we mean that V (P ) ⊆ W . Let G be a graph and let P = v 1 . . . v ℓ be a path on V (G). Let v := v 1 and u := v ℓ be the endpoints of P . Suppose v i ∈ N G (v) for some i = ℓ. Then, we can also consider the path P ′ = v i−1 v i−2 . . . vv i v i+1 . . . u in G ∪ P . We refer to the path P ′ as a rotation of P within G with fixed endpoint u and pivot v i . We call v i−1 v i the broken edge of the rotation.
Starting from P , we will consider successive rotations of P to obtain new paths, always leaving one of the endpoints of P fixed. We only consider rotations whose broken edges are edges in the original path P .
For any vertex x ∈ V (P ), let x − P,u and x + P,u denote the predecessor and successor of x along P , respectively (where P is oriented towards the fixed endpoint u). Similarly, given any set X ⊆ V (P ), we denote X + P,u := {x + P,u : x ∈ X} and X − P,u := {x − P,u : x ∈ X}. Let R G,P,u ⊆ V (P ) be the set of all vertices x ∈ V (P ) such that there exists a path P x in G ∪ P with endpoints u and x which can be obtained by taking successive rotations of P within G with fixed endpoint u. (As mentioned before, we only consider rotations whose broken edges are in P .) Whenever we consider a vertex x ∈ R G,P,u , the notation P x will be used to denote a path with endpoints x and u which can be obtained by the minimum number of rotations of P (whenever there is more than one choice for P x , we fix such a choice arbitrarily among all the possibilities). Let R 0 G,P,u := {v} and R t G,P,u be the set of vertices x ∈ R G,P,u such that P x is obtained by at most t rotations.
Given any set A ⊆ R G,P,u , we denote by R G,P,u (A) the union of A and the set of endpoints of all paths which are obtained via a single rotation of P a with u as a fixed endpoint, for any a ∈ A.
The following observation is well-known. We include the short proof in the appendix.
Lemma 4.1. Let G be a graph. Let P ′ be a path on V (G) and let P = v 1 . . . v ℓ be a longest path in G ∪ P ′ . Then, for all t ≥ 0 we have Next, we restrict ourselves to the random graph G n,p . Given a 'large' set A of endpoints obtainable via a 'small' number of successive rotations of a longest path P , we prove a lower bound on the number of endpoints obtainable from A via one further rotation.
For p ≥ C log n/n, the random graph G = G n,p a.a.s. satisfies the following. Let G ′ be a subgraph of G and P ′ be a path on V . Let P = v 1 . . . v ℓ be a longest path in G ′ ∪ P ′ . Then, for all A ⊆ R η log n G ′ ,P,v ℓ with |A| ≥ εn/100, we have that The proof of Lemma 4.2 is similar to (part of) the proof of Lemma 3.2 in [25]. For completeness, we include the details in the appendix.
We now combine the two previous results to give a lower bound on the number of endpoints which can be generated via successive rotations of a path P with one fixed endpoint.
Proof. Let u := v ℓ . Throughout this proof we write R t for R t G ′ ,P,u and R(A) := R G ′ ,P,u (A) for any A ⊆ R G ′ ,P,u . Let η be a number such that 1/C ≪ η ≪ ε. Condition on the event that the following holds for all v ∈ V : We also condition on the event that the assertions of Proposition 2.5 and Lemma 4.2 hold for G. By Lemmas 2.3 and 4.2 and Proposition 2.5, each of these events holds a.a.s. Note that (4.1) and the fact that H ∈ H ε n,p (G) imply that, for any set X ⊆ V with |X| ≥ εn/10, there exists a set X ′ ⊆ X with |X ′ | ≥ εn/20 and δ G ′ (X ′ ) ≥ min{|X|, n/2}p + εnp/2. (4.2) Note that, since P is a longest path in G ′ ∪P ′ , we have that N G ′ (x) ⊆ V (P ) for all x ∈ R G ′ ,P,u . We will consider successive rotations of P , keeping u fixed, to derive a lower bound on the number of distinct endpoints of different longest paths in G ′ ∪ P ′ with an endpoint u.
Definition 4.4. Let δ > 0. We say that a connected n-vertex graph G has property RE(δ) if one of the following holds for every path P on V (G): (i) there exists a path longer than P in the graph G ∪ P , (ii) there exists S P ⊆ V (G) with |S P | ≥ δn and a collection {T v : v ∈ S P } of subsets of V (G) with |T v | ≥ δn for all v ∈ S P satisfying the following: for all v ∈ S P and w ∈ T v , the graph G ∪ P contains a path Q between v and w with V (Q) = V (P ).
Proof. Recall that G a.a.s. satisfies the assertions of Proposition 2.5 and Lemma 4.3. We prove that G ′ satisfies RE(1/2 + ε/4) conditioned on this.
By Proposition 2.5(iii), G ′ is connected. Let P be any path on V . We may assume that G ′ ∪ P does not contain a path which is longer than P . Let one of the endpoints of P be u. By Lemma 4.3, there exists S P ⊆ V with |S P | ≥ (1/2+ε/4)n and such that, for every v ∈ S P , there exists a path Q v ⊆ G ′ ∪ P with endpoints u and v such that V (Q v ) = V (P ). For each path Q v we can fix v and apply Lemma 4.3 again to obtain a set T v ⊆ V such that |T v | ≥ (1/2 + ε/4)n and for every x ∈ T v there is a path Q xv ⊆ G ′ ∪ P from x to v with V (Q xv ) = V (P ). The result follows.
Definition 4.6. Let δ > 0 and let G 1 be a graph on n vertices with property RE(δ). We say that a graph G 2 with V (G 2 ) = V (G 1 ) complements G 1 if, for every path P on V (G 1 ), one of the following holds: (i) there exists a path longer than P in G 1 ∪ P , (ii) there exist sets S P and T v as in Definition 4.4 and vertices v ∈ S P and w ∈ T v such that vw is an edge of G 1 ∪ G 2 .
Finally, we state two lemmas which are used to complete the proof of Theorem 1.2. The first says that, given G = G n,p and H ∈ H ε n,p (G), the graph G \ H complements every 'small' subgraph of G which has property RE(1/2 + ε/4). The final lemma then says that G ′ actually contains some such 'small' graph as a subgraph. We include the proof of Lemma 4.8 in the appendix.
Lemma 4.8. For every 0 < ε < 1, there exist C, δ > 0 such that for p ≥ C log n/n we have that G = G n,p a.a.s. satisfies the following property: for any H ∈ H ε n,p (G), the graph G \ H complements all graphs R ⊆ G which satisfy RE(1/2 + ε/4) and have at most δn 2 p edges. Lemma 4.9. For all 0 < ε, δ ≤ 1, there exists C > 0 such that, for p ≥ C log n/n, the graph G = G n,p a.a.s. satisfies the following property. Let H ∈ H 2ε n,p (G). Then, G \ H contains a subgraph with at most δn 2 p edges satisfying RE(1/2 + ε/4).
Proof. Let 1/n ≪ 1/C ≪ ε, δ and 1/c < 1. Let p ′ := δp. We say that a graph F on V is good if it has at most n 2 p ′ = δn 2 p edges and, for all H ∈ H ε n,p ′ , the graph F \ H satisfies RE(1/2 + ε/4). Otherwise, we call it bad. Given any graph F on V , letF be the graph obtained from F by taking every edge of F independently with probability δ.
LetP be the measure associated with the experimentF . Let P total be the product measure obtained from considering the experiments yielding G n,p andĜ n,p (i.e. with respective measures P andP). Note that, by definition, the edge distribution ofĜ n,p is identical to that of G n,p ′ . It follows by Lemmas 2.2 and 4.5 that P total [Ĝ n,p is good] = P[G n,p ′ is good] = 1 − o(1).
Let F be the collection of all graphs F on V for whichP[F is good] ≥ 3/4. Since o(1) = P total [Ĝ n,p is bad] ≥ P[G n,p / ∈ F] P total [Ĝ n,p is bad | G n,p / ∈ F] ≥ P[G n,p / ∈ F]/4, we know that P[G n,p / ∈ F] = o(1) or, in other words, P[G n,p ∈ F] = 1 − o(1). Thus, from now on, we consider G = G n,p and condition on the event that G ∈ F.
The proof of Theorem 1.2 now follows from the previous results.
Proof of Theorem 1.2. Let 1/n ≪ 1/C ≪ δ ≪ ε. Condition on the assertions of Lemmas 4.8 and 4.9 holding with ε/2 instead of ε, which happens a.a.s. We will show that for any H ∈ H ε n,p (G), the graph G \ H is Hamiltonian.
Let H be a graph as above. By Lemma 4.9, there exists a subgraph G * of G \ H which has at most δn 2 p edges and satisfies property RE(1/2 + ε/8). By Lemma 4.8 we have that G \ H complements G * . Therefore, Proposition 4.7 implies that G \ H is Hamiltonian.