On resilience of connectivity in the evolution of random graphs

In this note we establish a resilience version of the classical hitting time result of Bollob\'{a}s and Thomason regarding connectivity. A graph $G$ is said to be $\alpha$-resilient with respect to a monotone increasing graph property $\mathcal{P}$ if for every spanning subgraph $H \subseteq G$ satisfying $\mathrm{deg}_{H}(v) \leq \alpha \mathrm{deg}_{G}(v)$ for all $v \in V(G)$, the graph $G - H$ still possesses $\mathcal{P}$. We show that for $m \geq (\tfrac{1}{6} + o(1)) n \log n$, the largest connected component of a random graph $G_{n, m}$ is $(\tfrac{1}{2} - o(1))$-resilient with respect to connectivity. Additionally, we show that in the process where in each step an edge is added randomly to the empty graph, as soon as the minimum degree reaches one the graph becomes resiliently connected. The result is optimal in the sense that the constants $1/6$ in the number of edges and $1/2$ in the resilience cannot be improved upon. We obtain similar results for $k$-connectivity.


Introduction
Random graph theory dates back to 1959 and two seminal papers of Erdős and Rényi [12] and Gilbert [15]. By now it has become a well-studied research area with applications in different fields. A more recent trend started with a paper of Sudakov and Vu [28] is to study the resilience of certain properties in random graphs, which has since attracted considerable attention cf. e.g. [1,2,3,9,20,26] and recent surveys [8,27]. In other words, knowing that a random graph behaves in a certain way, the question then becomes how robust that behaviour is with respect to modifications (e.g. deletion of the edges). Two of the most prominent examples of robustness of graphs can be seen in the results of Turán [29] and Dirac [11]. Namely, Turán's theorem measures how many edges one is allowed to remove from a complete graph K n before the graph becomes K r -free (global resilience), while Dirac's theorem measures how many edges touching each vertex one is allowed to remove before the graph ceases to be Hamiltonian (local resilience). The removal of all the edges incident to a vertex with minimum degree prevents the containment of any spanning structure (e.g. connectivity, Hamiltonicity, perfect matching, etc.) giving a trivial upper bound on the global resilience in these cases. In order to study robustness of such properties it is natural to turn the attention to the local resilience.
In a more formal setting, local resilience of a graph G with respect to a monotone increasing graph property P is defined as follows: Definition 1.1. Let G = (V, E) be a graph, P a monotone increasing graph property, and α ∈ [0, 1] a constant. We say that G is α-resilient with respect to P if for every spanning subgraph H ⊆ G satisfying deg H (v) ≤ α deg G (v) for all v ∈ V , we have G − H ∈ P.
Generally, being α-resilient means that an adversary cannot destroy property P by removing an arbitrary α-fraction of the edges incident to every vertex. In light of this, Dirac's theorem states that the complete graph K n is (1/2)-resilient with respect to Hamiltonicity. On the other hand, allowing the adversary to remove a bit more than a (1/2)-fraction of the edges incident to each vertex proves to be enough in order to destroy all Hamilton cycles, even disconnect the graph. There is a vast number of important results in extremal combinatorics which study 'resilience' of the complete graph and we refer the interested reader to e.g. [10,16,17,18,19] In this paper we show that one of the first results in random graph theory holds in a resilient fashion. A fundamental result of Erdős and Rényi [12] shows that if m = n 2 (log n + c n ), then with high probability 1 G n,m is connected if c n → ∞. Its strongest 'hitting time' version was shown much later by Bollobás and Thomason [6] and is the one which we consider here (see also Bollobás [5,Chapter 7]). For an integer n ∈ N we let {G i } define the random graph process as follows. Let G 0 be an empty graph on n vertices; at every step i ∈ {1, . . . , n 2 } let G i be obtained from G i−1 by choosing an edge e i / ∈ G i−1 uniformly at random and adding it to G i . This defines a sequence of nested graphs {G i } N i=0 , where N = n 2 and G 0 is an empty graph and G N a complete graph on n vertices.
A trivial necessary condition for a graph to be connected is that it has minimum degree at least one. It turns out that in the random graph process, this is also sufficient. In other words, the edge e i that makes the last isolated vertex disappear w.h.p. also makes the graph G i connected. Put into a more formal setting, for a monotone increasing graph property P and a random graph process {G i }, we define the hitting time with respect to P, and write τ P , as: One of the results of Bollobás and Thomason [6] proves that τ 1 = τ conn , where τ 1 and τ conn denote the hitting times of 'having minimum degree at least one' and of 'connectivity', respectively.
The hitting time statements about the random graph processes are in some sense the most precise results concerning random graphs one could hope for. Unsurprisingly, they are also the most difficult to analyse and only a handful of statements are known regarding some basic graph properties (such as connectivity, perfect matchings, Hamiltonicity, etc.).
Our contribution is to show that not only the graph is connected at the point τ 1 , but it is resiliently connected, that is it stays connected even after the adversary removes at most (1/2 − o(1))-fraction of the edges touching each vertex. This continues the line of research recently initiated by Nenadov, Steger, and the second author [25], of studying resilience of properties in random graph processes. We further obtain similar results with respect to k-connectivity.

Results
Our primary objective is to show the resilience of connectivity. However, instead of dealing with the random graph process {G i } directly, it is more convenient to work with the Erdős-Rényi random graph G n,m , namely a graph drawn uniformly at random among all graphs with n vertices and m edges. It is a well-known fact that for every m ∈ {1, . . . , n 2 } the graph G m has the same distribution as G n,m . Moreover, as long as m is above a certain value (which is the case for us), standard connections between the model G n,m and the binomial random graph model G n,p 2 lead to direct consequences for this case as well.
Our first result shows that as long as m is not too small, the largest connected component of G n,m , which we refer to as the giant, is resilient with respect to connectivity. The value of m is asymptotically optimal. Indeed, having m = 1−ε 6 n log n is enough for the existence of cherries 'attached' to the giant, i.e. two vertices of degree one with a common neighbour of degree three, which the adversary can easily disconnect by removing the edge from the degree three vertex which is not incident to a degree one vertex (see, [23]). Additionally, the constant 1/2 is optimal as we show in Proposition 3.4 at the end of Section 3.
The previous theorem immediately implies a hitting time result for resilience of connectivity. Theorem 1.3. Let ε > 0 be a constant. Consider the random graph process {G i } and let τ 1 = min{m : δ(G m ) ≥ 1} denote the step in which the last isolated vertex disappears. Then w.h.p. we have that G τ 1 is (1/2 − ε)-resilient with respect to connectivity.
We now turn our attention to k-connectivity. Recall, a graph is said to be k-connected if the removal of at most k − 1 vertices does not disconnect it. For the random graph process {G i }, let us define τ k = min{m : δ(G m ) ≥ k} and τ k−conn = min{m : G m is k-connected}, denoting the point at which δ(G m ) ≥ k and at which G m becomes k-connected, respectively. Erdős and Rényi [13] were the first to show that for m = n 2 (log n + (k − 1) log log n + c n ) w.h.p. G n,m is k-connected if c n → ∞, which also coincides with the threshold for G n,m to have minimum degree k. This result was later strengthened by Bollobás and Thomason [6] who proved the hitting time statement, that is w.h.p. τ k = τ k−conn . Therefore, the trivial necessary condition for k-connectivity-having minimum degree at least k-turns out to be sufficient as well.
In order to state our second result regarding k-connectivity, we need a slightly different notion of resilience. In the case of 1-connectivity (or simply connectivity) and 2-connectivity, (1/2− o(1))resilience is not enough for the adversary to make the minimum degree of the graph drop below one and two, respectively. However, as soon as k ≥ 3, the adversary would be allowed to remove an edge incident to a degree three vertex, and could easily prevent 3-connectivity. We go around this fact by slightly restricting the power of the adversary with the following definition. Definition 1.4 ((α, k)-resilience). Let G = (V, E) be a graph, P a monotone increasing graph property, α ∈ [0, 1] a constant, and k ≥ 1 an integer. We say that G is (α, k)-resilient with respect to P if for every spanning subgraph As observed above, the main obstruction for not having k-connectivity in the beginning of the process is the existence of vertices with degree smaller than k. However, something can still be said for such sparse(r) graphs. For an integer k ≥ 2 and a graph G we define the k-core of G to be the graph obtained by successively removing vertices of degree less than k from G. With this in mind we are ready to state our second result. Theorem 1.5. Let k ≥ 2 be an integer, ε > 0 a constant, and consider the random graph process {G i }. Then w.h.p. for every m ≥ 1+ε 6 n log n we have that the k-core of G m is (1/2−ε, k)-resilient with respect to k-connectivity.
Even though the value m ≥ n log n/6 is optimal for connectivity, it is conceivable that the kcore is resilient with respect to k-connectivity much earlier in the process, that is as soon as it is k-connected, which happens roughly for m = cn as shown by Bollobás [4]. From a recent result of Montgomery [24] one deduces that the k-core is (1/2 − o(1))-resilient with respect to 2-connectivity, but the constant k is much larger than two, and furthermore, no conclusions can be drawn for k-connectivity. We leave this as a question for further research.
Similarly as before we immediately obtain a hitting time version of the result. Theorem 1.6. Let k ≥ 2 be an integer and ε > 0 a constant. Consider the random graph process {G i } and let τ k = min{m : δ(G m ) ≥ k} denote the step in which the last vertex of degree As a direct corollary we also get a statement about the classic notion of (1/2 − o(1))-resilience for k-connectivity. Corollary 1.7. Let k ≥ 2 be an integer and ε > 0 a constant. Consider the random graph

Notation and definitions
Our graph theoretic notation follows standard textbooks (see, e.g. [7]). In particular, given a graph G we denote by V (G) and E(G) the set of its vertices and edges, respectively, and by v(G) and e(G) their sizes. For subsets of vertices X, Y ⊆ V (G), G[X] stands for the subgraph of G induced by X, G[X, Y ] for the bipartite subgraph with biparition (X, Y ), and E G (X, Y ) denotes the set of edges between X and Y in G, i.e. E G (X, Y ) := {{v, w} : v ∈ X, w ∈ Y, and {v, w} ∈ E(G)} and e G (X, Y ) denotes its size. For short, we write E G (X) := E G (X, X) and e G (X) for its size. Furthermore, we write N G (X) := {v ∈ V (G) : ∃u ∈ X such that {v, u} ∈ E(G)} for the neighbourhood of X in G. Given a vertex v ∈ V (G) we abbreviate N G ({v}) to N G (v) and let deg G (v) be the size of its neighbourhood, that is the degree of v in G. We use δ(G) for the minimum degree of G. For ℓ ∈ N and a vertex v ∈ V (G), we define the ℓ-neighbourhood of v as the set of all vertices which lie at distance at most ℓ from v, and write N ℓ G (v), excluding v itself. We omit the subscript G whenever it is clear from the context which graph we refer to.
For an integer n we write [n] := {1, . . . , n} and for x, y, ε ∈ R, we write x ∈ (1 ± ε)y to denote (1 − ε)y ≤ x ≤ (1 + ε)y. Throughout, we use the natural logarithm log x = log e x. Ceilings and floors are omitted whenever they are not essential. We make use of the standard asymptotic notation o, ω, O, and Ω. Lastly, we use subscripts with constants such as C 3.1 to indicate that it is the function defined in the statement of Claim/Lemma/Proposition/Theorem 3.1.

Probabilistic tools
First is a simple but useful bound on the binomially distributed random variable.
Throughout we often make use of the standard Chernoff bounds estimate for deviation of a binomially distributed random variable Bin(n, p) with parameters n and p from its mean (see, e.g. [14]) Lemma 2.2 (Chernoff bounds). Let X ∼ Bin(n, p) and let µ := E[X]. Then for all δ ∈ (0, 1): The inequalities above remain true if X has hypergeometric distribution with the same mean.

Random graphs
Although our main results concern the random graph process, in the proof we heavily rely on the properties of the binomial random graph G n,p . In this subsection we collect several useful (and mostly well-known) properties of such graphs. The first one gives a bound on the maximum number of edges certain subsets can have.
Proof. Both assertions follow from standard arguments, but nevertheless we spell them out for completeness. Let us first prove (i). By a union bound argument, the probability that there is a set X ⊆ V (G) which violates the property is at most The biggest obstacle in working with (very) sparse random graphs, that is for values of p close to the threshold of connectivity, is that one cannot guarantee that even the degree of every vertex is w.h.p. concentrated around its expectation. Hence, the vertices whose degrees deviate significantly from the average require some special care. We introduce two classes of such vertices in the following definition.
Definition 2.4. For δ, p ∈ [0, 1] and a graph G with n vertices, we define We refer to the vertices in TINY p,δ (G) as tiny and to the vertices in ATYP p,δ (G) as atypical.
Note that if the graph G in the above definition is a random graph G n,p and p is roughly such that np is the average degree of a vertex then one can think of TINY as a set of vertices whose degree is much smaller than the average, and ATYP as a set of vertices whose degree is even slightly away from the average. Let us point out that if p ≥ (1 + ε) log n/n and δ is small enough (depending on ε), then w.h.p. G n,p contains no tiny vertices, and similarly if p ≥ C log n/n for large enough C (depending on δ), then w.h.p. G n,p contains no atypical vertices.
We briefly discuss the fact that with the previous definition at hand we could use a variant of [25,Definition 2.4] in order to show marginally stronger statements than the ones from Theorem 1.3 and Theorem 1.6. Namely, one could allow an adversary to remove all but one (or two) edges incident to tiny vertices and all but a large constant number of edges incident to atypical vertices. However, just using (1/2 − o(1), δ t , 1, δ a , K a )-resilience for connectivity would not suffice for the following reason: for m ≤ (1/4 − ε)n log n w.h.p. there exists a path of length two 'attached' to the giant of G m , which then an adversary could disconnect by removing the edge that 'attaches' it to the giant. Note that the removal of such an edge is not possible under the definition of (1/2 − o(1))-resilience and hence we would need a slight modification in the definition from [25]. Lastly, (1/2 − o(1), δ t , k, δ a , K a )-resilience would suffice for k-connectivity for every k ≥ 2. In conclusion, we believe that such a 'complication' would greatly reduce the readability of our paper, unnecessarily distract the reader from the main point, and is as such not worth pursuing.
As their name indicates, the atypical vertices are 'rare' and do not occupy a significant fraction of the graph.

Proof. For a fixed vertex v ∈ V (G) by Chernoff bounds we have
for some γ > 0. Therefore, the expected size of the set ATYP p,δ (G) is n 1−γ and Markov's inequality shows that w.h.p. |ATYP p,δ (G)| ≤ n/ log n.

The proof
Recall that we are trying to prove that the random graph process {G i } is typically such that at the point τ conn the graph G τconn is resilient with respect to connectivity. Before diving into the details we give a brief outline of the ideas used.
We follow the exact path paved by the proof of [25, Proposition 3.1]. The most natural thing one could try is to show that for a fixed m ≥ n log n/6 the giant of G m is w.h.p. resilient with respect to connectivity and apply a union bound over all such m. Unfortunately, this approach would fail as this probability is proportional to the average degree and is roughly 1 − e −α·2m/n , for a small constant α > 0-clearly not enough for a union bound over all values of m of order n log n. Instead, we group graphs into batches of size εn log n and show that w.h.p. all graphs in a batch satisfy the property simultaneously. This allows us to union bound over only a constant number of batches in order to cover all values of m up to Cn log n, for sufficiently large constant C. Choosing C to be large enough, we may then union bound over all remaining values of m individually, as now for each fixed one the statement holds with probability at least 1 − o(n −3 ).
For a cleaner exposition, we generate the graph G m with the help of a binomial random graph G n,p , instead of doing it from scratch. Namely, we sample the graphs G − ∼ G n,p 0 and G n,p ′ where the values of p 0 and p ′ are such that G − w.h.p. has at most some m 0 edges and G + = G − ∪ G n,p ′ has at least (1 + ε/6)m 0 edges. Taking now a permutation π of the edges of G n,p ′ uniformly at random we may generate each G m as a union of G − and the first m − e(G − ) edges given by π.
Crucially, the properties of the graphs G − and G + that we make use of are such that all graphs 'squeezed' in between them also satisfy them. The most important one concerns tiny and atypical vertices and states that not many of them can be clumped together. Actually, an even stronger statement is true: adding an additional ε-fraction of random edges to G − in order to obtain G + does not make tiny and atypical vertices of G − get more clumped. The next lemma, which is a slightly simplified version of [25, Lemma 2.7], captures this precisely. We omit the proof.
. Then: Lastly, in order for the whole strategy to work in the resilience setting we keep only the edges of G m existing in G − , but allow the adversary to remove with respect to the degrees in G m . This makes the conditions strictly weaker. Proposition 3.2. Let k ≥ 1 be an integer. For every ε > 0 and integer m 0 ≥ 1+ε 6 n log n the random graph process {G i } w.h.p. has the following property: for every integer m 0 ≤ m ≤ (1 + ε/6)m 0 the giant of G m is (1/2 − ε)-resilient with respect to connectivity. Furthermore, if k ≥ 2, the k-core of G m is (1/2 − ε, k)-resilient with respect to k-connectivity.
For the rest of the proof consider some m 0 ≤ m ≤ (1+ε/6)m 0 . As the proof for both connectivity and k-connectivity (for k ≥ 2) is identical, we treat them together and think of the graph we work with as the giant of G m in the former and as the k-core of G m in the latter. With this in mind, let G ⊆ G m be the giant of G m , respectively a subgraph obtained by iteratively removing all vertices of degree less that k from G m for k ≥ 2. Let V = V (G) denote the vertex set of G, and let TINY and ATYP be sets of vertices defined as: It is a well-known fact that for m ≥ n log n/6 the size of the giant (resp. the k-core) of G m is at least (1 − o(1))n (cf. [14,21,22]). If we show that for every graph H whose vertex degrees fulfil the graph G − H is k-connected, then G is (1/2 − ε, k)-resilient with respect to k-connectivity.
First, we list a series of properties that the graph G satisfies, which we subsequently show are sufficient for proving the k-connectivity of G − H: with v(C) = 3 contains at most one vertex from TINY, (C5) |ATYP| ≤ n log n . We show that the properties hold in G + and hence in every subgraph G ⊆ G + . Properties (C1) and (C2) are given by Lemma 2.3 applied to G + with ε/4 as ε and p 1 as p, since p 1 ≥ p 0 > log n/(3n). Properties (C3) and (C4) hold by our choice of δ and L, and by Lemma 3.1 applied with ε/2 as ε, p 0 and p ′ , since p 0 ≥ (1 + ε/2) log n/(3n). Lastly, (C5) holds by Lemma 2.5.
Consider a graph H on the same vertex set V as G which satisfies (1), and let G ′ := G − H. We prove the k-connectivity of G ′ by showing that for every S ⊆ V of size |S| ≤ k − 1, the neighbourhood of every X ⊆ V \S is not completely contained in X in the graph Without loss of generality, we only consider |X| ≤ |V |/2. Assuming towards a contradiction that all edges incident to the vertices belonging to X have both endpoints in X, gives From (1) we see that all vertices v in G ′ satisfy where the last part follows from the fact that every vertex v ∈ V \ ATYP has degree at least (1 − δ)np 0 in G and thus making use of our choice of δ and p 1 in the last inequality.
We consider two cases depending on the size of X: (I) |X| ≤ n 1000 and (II) |X| > n 1000 . Let us look at (I) first. The most critical point is to show that a significant part of the vertices in X are actually not in TINY. The next claim establishes precisely that. Proof. Observe first that there cannot exist a path of length two in the induced subgraph G ′′ [X ∩ TINY]. Indeed, if such a path exists, as it was a part of the giant G initially, it follows that there was a vertex v ∈ V with three tiny vertices (the ones on the path) in N 3 G (v)-a contradiction to (C3). Consequently, the graph G ′′ [X ∩ TINY] consists only of isolated vertices and edges. Let X E and X V be the partition of X ∩ TINY into a matching and an independent set and let If k = 1 then as every edge {u, v} ∈ X E was initially a part of the giant G, we have |N E | ≥ |X E | due to (C3) and (3). Additionally, if k ≥ 2, by (3) together with properties (C3) and (C4) one easily derives Similarly, as no three vertices from X V can have a common neighbour in X \ TINY by (C3), we deduce |N V | ≥ ⌈|X V |/2⌉. Indeed, every vertex has at least k neighbours in X \ TINY and at most k − 1 are removed due to the removal of S; moreover, at most one other vertex in X V can share this remaining neighbour due to (C3).
Clearly now, if the number of vertices in X \ TINY is strictly smaller than |N E | + |N V | by the pigeonhole principle there is a vertex v ∈ X \ TINY which has at least three vertices from X ∩ TINY in N 3 G ′′ (v), again a contradiction to (C3). The claim readily follows.
Next, we show that |X typ | > |X atyp |. Assume towards contradiction that this is not the case. By the minimum degree assumption (3) we have where the second inequality follows from property (C3) and the last from our assumption that |X atyp | ≥ |X typ | and np 0 = ω(1). Thus, a simple averaging argument shows that there exists a vertex u ∈ X typ with deg G ′′ (u, X atyp ) > L, which is a contradiction to (C3). Together with Claim 3.3 we conclude |X typ | ≥ |X|/6. Therefore, we have where we again make use of (3). On the other hand, by (C1) we also have e G ′′ (X) ≤ |X|np 1 /1000, hence the assumption N G ′′ (X) \ X = ∅ is wrong.
In conclusion, there is no connected component of G ′′ of size at most |V |/2, which completes the proof of the proposition.
Having Proposition 3.2 at hand we proceed to complete the proof of the main result. For convenience of the reader we restate the theorem. Proof. One can easily see that the conclusion of Proposition 3.2 holds with probability 1 − e −α·C log n for every fixed m ≥ Cn log n, some α > 0, and for sufficiently large C depending on ε. Note that it is also possible to draw the same conclusion from the result of Lee and Sudakov [20,Theorem 1.1] as connectivity is a necessary condition for Hamiltonicity. Therefore, a union bound over all m ≥ Cn log n shows that with probability at least 1 − n 2 · e −αC log n = 1 − o(1), every G m is resilient with respect to connectivity.
For the smaller values of m, consider intervals of the form [ 1+iε 6 n log n, 1+(i+1)ε 6 n log n), for i ∈ 1, . . . , C ε , where C ε is such that the last interval contains Cn log n. For each fixed interval the conclusion of Proposition 3.2 holds with probability 1− o(1), thus a union bound over constantly many intervals shows that it w.h.p. holds for all intervals simultaneously. This concludes the proof.
The proof of Theorem 1.5 is analogous. This immediately implies Theorem 1.3 and Theorem 1.6.
We conclude by showing that if an adversary is allowed to remove slightly more than a (1/2)fraction of the edges incident to each vertex, then w.h.p. it is possible to disconnect the giant of G m . For simplicity we only show it for one fixed value of m. In order to capture all values of m the statement and the proof would go along the lines of Proposition 3.2.
Proposition 3.4. For every ε > 0 and integer m ≥ 1+ε 6 n log n the random graph process {G i } w.h.p. is such that the giant of G m is not (1/2 + ε)-resilient with respect to connectivity.
Proof. Let G be the giant of G m and V its vertex set, take δ = min{ε/32, δ 3.1 (ε/2)} and p = e(G)/ v(G) 2 , and further assume that (C3) holds for G. Suppose there exists a partition V = A ∪ B such that the following property is satisfied: Then by taking H to be a subgraph consisting of all edges in G[A, B] the graph G − H is not connected. In the remainder we show that such a partition indeed exists.
Consider first an arbitrary equipartition V = A ′ ∪ B ′ . Let TINY := TINY p,δ (G) ∩ V as before, and let D be defined as: In other words, D is the set of all vertices which have more than the expected degree in G[A ′ , B ′ ] or less than the expected degree in G − G[A ′ , B ′ ]. Note that, trivially, TINY ⊆ D. We claim that such vertices cannot be bundled together. Proof. We show that if T ⊆ G is a tree with L ′ ≤ v(T ) ≤ 2L ′ vertices then it contains at most L ′ − 1 vertices from D, which is sufficient for the claim to hold. Clearly, if for a vertex v ∈ V (G) there are L ′ vertices in N G (v) ∩ D then taking a tree containing v and every such vertex u together with the edge {v, u} contradicts the former.
Let T ⊆ K n be a tree with L ′ ≤ v(T ) ≤ 2L ′ and S ⊆ V (T ) a set of size exactly L ′ . Let E T denote the event that T ⊆ G and E T,S that every v ∈ S satisfies either |N G−G[A ′ ,B ′ ] (v) \ V (T )| < (1/2 − δ/2)np or |N G[A ′ ,B ′ ] (v) \ V (T )| > (1/2 + δ/2)np. By Chernoff bounds we have that the probability that a fixed vertex v ∈ S satisfies this is at most Pr Bin n 2 − v(T ), p < (1/2 − δ/2)np + Pr Bin n 2 , p > (1/2 + δ/2)np ≤ e −γnp , for some γ > 0 depending on δ (and thus ε). As these events are independent for different vertices v, u ∈ S, the probability that E T,S holds is bounded by Note that Pr[E T ] = p v(T )−1 . Hence, a simple union bound over all pairs (T, S) shows that the probability that some E T ∧ E T,S happens is at most As np = Ω(log n) and n t p t−1 · e −L ′ γnp is decreasing in p, this finally implies Now, let us look at tiny vertices. If after the end of the process there is no vertex v ∈ TINY which has deg G[A,B] (v) > deg G (v)/2, we are done. Hence, assume w.l.o.g. that v ∈ A is such a vertex. In particular, this implies that v had a neighbour u ∈ TINY added to B after v itself. Since u ∈ B, it implies that there exists w ∈ N G (u, B). Consequently, due to (C3), none of the neighbours of v in A belong to TINY and we may move v to B without harming the degree of any vertex significantly (it changes by at most one for non-tiny vertices, which is negligible compared to their degree). Moreover, by doing this we cannot make additional vertices v ′ ∈ TINY have deg G[A,B] (v ′ ) > deg G (v ′ )/2 and thus the rearranging process eventually stops with all vertices satisfying (⋆). This completes the proof.