Resilience of Perfect Matchings and Hamiltonicity in Random Graph Processes
RResilience of Perfect Matchings and Hamiltonicity in RandomGraph Processes
Rajko Nenadov ∗ Angelika Steger † Miloˇs Truji´c † , ‡ Abstract
Let { G i } be the random graph process: starting with an empty graph G with n vertices,in every step i ≥ G i is formed by taking an edge chosen uniformly at randomamong the non-existing ones and adding it to the graph G i − . The classical ‘hitting-time’result of Ajtai, Koml´os, and Szemer´edi, and independently Bollob´as, states that asymptoti-cally almost surely the graph becomes Hamiltonian as soon as the minimum degree reaches 2,that is if δ ( G i ) ≥ G i is Hamiltonian. We establish a resilience version of this result.In particular, we show that the random graph process almost surely creates a sequence ofgraphs such that for m ≥ ( + o (1)) n log n edges, the 2-core of the graph G m remains Hamil-tonian even after an adversary removes ( − o (1))-fraction of the edges incident to everyvertex. A similar result is obtained for perfect matchings. The theory of random graphs originated in 1959 with the two seminal papers by Erd˝os andR´enyi [13] and Gilbert [16]. It is now a well-established research area with many applicationsin theoretical computer science, statistical physics, and other branches of mathematics, cf. [6,14, 19]. Typical questions in random graph theory traditionally concern the existence of certain(sub)structures. A more recent trend, introduced by Sudakov and Vu [33], is the study of their resilience properties. Formally, resilience of a graph G with respect to some property P isdefined as follows: Definition 1.1.
Let G = ( V, E ) be a graph. We say that G is α -resilient with respect to theproperty P , for some α ∈ [0 , H ⊆ G such that deg H ( v ) ≤ α deg G ( v ) for every v ∈ V , we have G − H ∈ P .In other words, α -resilience implies that the property P cannot be destroyed even if an adversaryis allowed to remove an (arbitrary) α -fraction of all edges incident to each vertex. For instance,Dirac’s celebrated theorem [11] states that K n is (1 / n ∈ N and consider ∗ School of Mathematical Sciences, Monash University, VIC 3800, AustraliaEmail: [email protected] † Institute of Theoretical Computer Science, ETH Z¨urich, 8092 Z¨urich, SwitzerlandEmail: { steger | mtrujic } @inf.ethz.ch ‡ author was supported by grant no. 200021 169242 of the Swiss National Science Foundation. a r X i v : . [ m a t h . C O ] A ug he following process: set G to be an empty graph with n vertices and for each i ∈ { , . . . , (cid:0) n (cid:1) } form the graph G i by choosing an edge e i / ∈ G i − uniformly at random and adding it to G i − .In this way we obtain a sequence of nested graphs { G i } Ni =0 for N = (cid:0) n (cid:1) , where G is an emptygraph and G N is a complete graph. It is an easy exercise to show that for every m ∈ { , . . . , N } the graph G m has the same distribution as the Erd˝os-R´enyi random graph G n,m , a graph chosenuniformly at random among all labelled graphs with n vertices and exactly m edges. A necessarycondition for a graph to contain a perfect matching is that the minimum degree is at least 1(assuming that n is even). Similarly, having minimum degree at least 2 is a necessary conditionto be Hamiltonian. The aforementioned results show that, perhaps surprisingly, these conditionsare also sufficient: as soon as δ ( G m ) ≥ G m contains a perfect matching, and as soonas δ ( G m ) ≥ . Thiskind of results are usually referred to as hitting-time results.In this paper we show that, in fact, as soon as the minimum degree reaches 1 the graph doesnot only contain a perfect matching, but it does so robustly —meaning it is (1 / − o (1))-resilientwith respect to this property. Similarly, as soon as the minimum degree reaches 2 the graph G m becomes robustly Hamiltonian, that is (1 / − o (1))-resilient with respect to being Hamiltonian.To see why the parameter 1 / G contains a partition V ∪ V such that every vertex in one part has at least half of its neighboursin the other. In the case when such a partition is not balanced, deleting all the edges within V and V clearly prevents the resulting graph from having either a perfect matching or a Hamiltoncycle. Otherwise, both parts have size n/ G is arandom graph with m (cid:29) n edges, which will be the case here, at least one vertex in V alsohas roughly half of its neighbours in V . Moving such a vertex to V results in an unbalancedpartition, and the rest of the argument is the same as before.Even though the study of resilience properties of random graphs has attracted considerableattention over the last years, cf. e.g. [2, 8, 18, 23] and the recent survey [32], this is the firstresult which determines precisely when the random graph process becomes resilient with respectto some well-studied property. The classical hitting-time result for perfect matchings by Bollob´as and Thomason [7], and apreviously obtained result by Erd˝os and R´enyi [12] which establishes the correspondence betweenminimum degree one and perfect matchings in a slightly weaker form, follow quite easily fromwell-known sufficient conditions for the existence of perfect matchings. On the other hand, thehitting time result for Hamiltonicity by Ajtai, Koml´os, and Szemer´edi [1] and Bollob´as [5] issignificantly more intricate and builds on a result of P´osa [31] and subsequent refinements.The situation is similar for our results. While Hamiltonicity is more interesting and challenging(and our main result), we use the case of perfect matchings to demonstrate some of our mainideas and the general approach.
Perfect matchings.
Our first result shows a bit more than discussed earlier: as long as thenumber of added edges is not too small, removing isolated vertices, an obvious obstacle inobtaining a perfect matching, results in a graph which is resilient with respect to containing aperfect matching . An event is said to hold asymptotically almost surely (a.a.s. for short) if the probability that it holds ap-proaches 1 as n → ∞ . In order to avoid having to distinguish between an odd and an even number of vertices of the given graph, weuse the term ‘perfect matching’ to indicate that the graph contains a matching that covers all but at most one of heorem 1.2. Let ε > be a constant and consider the random graph process { G i } . Thena.a.s. for every m ≥ ε n log n we have that the graph obtained from G m by deleting all isolatedvertices is (1 / − ε ) -resilient with respect to containing a perfect matching. This improves a result of Sudakov and Vu [33] who showed that a random graph G n,m is (1 / − ε )-resilient with respect to containing a perfect matching if m ≥ Cn log n , for some large constant C ( ε ) > m in Theorem 1.2 is asymptotically optimal. The reasonbeing that for m = − ε n log n there exist many cherries (see, e.g. [7]), pairs of vertices of degree1 which have a common neighbour. Clearly, for each cherry only one vertex can be part ofa matching and a perfect matching thus cannot exist. The standard non-resilience version ofTheorem 1.2 with somewhat more precise lower bound on m was established in [7].In fact, we obtain a slightly stronger resilience statement. That is, even if we allow the adversaryto delete a graph H that contains almost all the edges incident to ‘atypical’ vertices (verticeswhose degree deviates significantly from the average degree), the resulting graph still containsa perfect matching. The precise statement can be found in Section 3.Theorem 1.2 also immediately implies a resilience version of the hitting-time result for perfectmatchings. Theorem 1.3.
Let ε > be a constant. Consider the random graph process { G i } with an evennumber of vertices n and let m = min { m : δ ( G m ) = 1 } denote the step in which the last isolatedvertex disappears. Then a.a.s. we have that G m is (1 / − ε ) -resilient with respect to containinga perfect matching. Hamiltonicity.
Prior to proving that the point when δ ( G m ) becomes 2 coincides with thepoint when G m is Hamiltonian, Koml´os and Szemer´edi [21] showed that if m = n (log n +log log n + c n ) then G n,m contains a Hamilton cycle if c n → ∞ . On the other hand, if c n → −∞ then it is a.a.s. not Hamiltonian (see [21]), precisely because of the existence of a vertex withdegree less than 2. However, something can still be said for such a sparse graph, or ratherits 2 -core . Given a graph G we define the 2 -core of G to be the graph obtained from G bysuccessively removing vertices of degree at most 1. In other words, the 2-core of a graph G is itslargest subgraph which satisfies the necessary minimum degree condition for being Hamiltonian.(cid:32)Luczak [28] showed that, in the case of a random graph G n,m , such subgraphs are indeedHamiltonian as long as m ≥ ε n log n . Here we extend this by showing that, starting fromthat point, the 2-core of every graph G m is resilient with respect to containing a Hamiltoncycle. Moreover, the bound on m is asymptotically optimal, as a.a.s. the 2-core of G n,m is notHamiltonian for m ≤ − ε n log n (see [28]). Theorem 1.4.
Let ε > be a constant and consider the random graph process { G i } . Thena.a.s. for every m ≥ ε n log n we have that the -core of G m is (1 / − ε ) -resilient with respectto being Hamiltonian. Similarly to the case of perfect matchings we actually prove a slightly stronger resilience re-sult that allows deletion of almost all edges incident to ‘atypical’ vertices, cf. Section 4 andTheorem 4.8 for details.The question of resilience of random graphs with respect to Hamiltonicity has attracted consid-erable attention in the last years. Sudakov and Vu [33] showed that G n,m is (1 / − o (1))-resilientwith respect to Hamiltonicity for m ≥ n log n , and conjectured that already m (cid:29) n log n shouldsuffice. The lower bound on m was gradually improved in a series of paper: Frieze and Krivele-vich [15] showed that G n,m is α -resilient for p ≥ Kn log n , for some small α > K ; Ben-Shimon, Krivelevich, and Sudakov [3] showed (1 / − o (1))-resilience and soon after its vertices. / − o (1))-resilience [4] for m ≥ ε n log n ; finally, Lee and Sudakov [27] showedthe optimal (1 / − o (1))-resilience for m ≥ Kn log n .While the last result settles the original conjecture of Sudakov and Vu, the gap between the valueof m for which G n,m is typically Hamiltonian and for which we know is (1 / − o (1))-resilient withrespect to Hamiltonicity remains. A simple corollary of Theorem 1.4 closes this gap by showingthat the two coincide. More precisely, we obtain that as soon as δ ( G m ) ≥ G m isresilient with respect to Hamiltonicity. This provides a resilience version of the hitting-timeresult of Ajtai, Koml´os, and Szemer´edi [1] and Bollob´as [5]. Theorem 1.5.
Let ε > be a constant. Consider the random graph process { G i } and let m = min { m : δ ( G m ) = 2 } denote the step in which the last vertex of degree at most onedisappears. Then a.a.s. we have that G m is (1 / − ε ) -resilient with respect to being Hamiltonian. We note that Theorem 1.5 (but not 1.4) was proven independently also by Montgomery [30]using a different approach.
Structure of the paper.
In the next section we give some necessary definitions and toolsused throughout the paper. In addition to providing standard facts, we introduce the notionof tiny and atypical vertices and a strengthened notion of α -resilience. In Section 3 we prove(a strengthened version) of Theorem 1.2 that is based on this new version of α -resilience. Eventhough the proof strategy is simple on a high level, it requires many intricate details related totiny and atypical vertices. In Section 4 we use some of these ideas and combine them with theP´osa’s rotation-extension technique to prove a strengthened version of Theorem 1.4. Our graph theoretic notation is standard and follows the one from [10]. In particular, givena graph G and (not necessarily disjoint) subsets X, Y ⊆ V ( G ), we denote by e G ( X, Y ) thenumber of edges of G with one endpoint in X and the other in Y . Note that every edge whichlies in the intersection of X and Y is counted twice. We denote by e G ( X ) the number of edgeswith both endpoints in X . Furthermore, given a vertex v and an integer (cid:96) ≥
1, we denote by N G ( v ) its set of neighbours and by N (cid:96)G ( v ) the set of all vertices which are at distance at most (cid:96) from v , excluding v . Given graphs G and H with the same vertex set, we define G − H asthe graph with vertex set V ( G ) and edge set E ( G ) \ E ( H ). Moreover, for a graph G and aset of edges E on the same vertex set, we write G + E to denote the graph on the vertex set V ( G ) and edge set E ( G ) ∪ E . For a positive integer n and a function 0 ≤ p := p ( n ) ≤ G n,p denote the probability space of graphs with vertex set [ n ] = { , . . . , n } where each pairof vertices forms an edge of G ∼ G n,p with probability p , independently of all other pairs. Wemake use of the standard asymptotic notation o, O, ω , and Ω. In addition, we write O ε or Ω ε to emphasise that the hidden constant depends on a parameter ε . For two functions a and b we write a (cid:28) b to indicate a = o ( b ) and a (cid:29) b to indicate a = ω ( b ). In all occurrenceslog denotes the natural logarithm. We mostly suppress floors and ceilings whenever they arenot crucial. Given ε, x, y ∈ R , we write x ∈ (1 ± ε ) y to denote (1 − ε ) y ≤ x ≤ (1 + ε ) y .Finally, we use subscripts with constants such as C . to indicate that C . is a constant givenby Claim/Lemma/Proposition/Theorem 2.5.Throughout the paper we make use of the following standard estimate on tail probabilities of abinomial random variable Bin( n, p ) with parameters n and p , see e.g. [14]. Lemma 2.1 (Chernoff bounds) . Let X ∼ Bin ( n, p ) and let µ := E [ X ] . Then for all < δ < : • Pr[ X ≥ (1 + δ ) µ ] ≤ e − δ µ , and Pr[ X ≤ (1 − δ ) µ ] ≤ e − δ µ .Remark . This result remains true if X has a hypergeometric distribution, cf. [19, Theorem2.10]. Even though our main results concern the random graph process rather than the binomialrandom graph G n,p , in the course of the proof we extensively rely on some of its properties. Forsmall values of p , in particular those which give rise to a number of edges as in Theorem 1.2 andTheorem 1.4, a random graph G n,p contains many vertices whose degrees deviate significantlyfrom the average degree. In our proofs those vertices require special attention. The followingdefinition captures them formally. Definition 2.2.
Given δ, p ∈ [0 ,
1] and a graph G with n vertices, we define the following setsof vertices: TINY p,δ ( G ) = { v ∈ V ( G ) : deg G ( v ) < δnp } , ATYP p,δ ( G ) = { v ∈ V ( G ) : deg G ( v ) / ∈ (1 ± δ ) np } . 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 in the above definition G is an arbitrary graph, not necessarily a G n,p . This is thereason why p appears in the index as well. In our applications p is usually chosen such that np is roughly (but not necessarily exactly) the average degree of G . Thus, as the names indicate,tiny vertices have a degree significantly smaller than the average degree and atypical ones arebounded away from the average by a small constant factor. If G ∼ G n,p then both of thesesets become empty as soon as p is large enough: for p ≥ (1 + ε ) log n/n there are a.a.s. no tinyvertices for sufficiently small δ (depending on ε ), and similarly for p ≥ C log n/n there are a.a.s.no atypical vertices if C is sufficiently large (depending on δ ).With this definition in mind we refine the notion of α -resilience. Definition 2.3.
Given a graph G = ( V, E ), a graph property P , constants α, δ t , δ a ∈ [0 , K t , K a ∈ N , we say that G is ( α, δ t , K t , δ a , K a ) -resilient with respect to P if for everyspanning subgraph H ⊆ G such thatdeg H ( v ) ≤ deg G ( v ) − K t , if v ∈ TINY p,δ t ( G ) , deg G ( v ) − K a , if v ∈ ATYP p,δ a ( G ) \ TINY p,δ t ( G ) ,α deg G ( v ) , otherwise , for every v ∈ V , where p = | E | / (cid:0) | V | (cid:1) , we have G − H ∈ P .Under some natural conditions, one easily sees that ( α, δ t , K t , δ a , K a )-resilience implies α -resilience. Lemma 2.4.
Let G = ( V, E ) be a graph with n vertices and α, δ t , δ a ∈ [0 , and K t , K a ∈ N constants such that (1 − α ) δ t np ≥ K a ≥ K t , where p = | E | / (cid:0) n (cid:1) . If the minimum degree d of G is such that d − (cid:98) αd (cid:99) ≥ K t , then ( α, δ t , K t , δ a , K a ) -resilience of G implies α -resilience of G . In particular, in our applications we have that K t is either 1 (for perfect matchings) or 2 (forHamilton cycles) and K a is a (large) constant. A random graph G n,p with p (cid:29) /n then a.a.s.trivially fulfils the first condition in the lemma above. In addition, for α = 1 / − ε , the secondcondition is also satisfied if we constrain G n,p to the subgraph induced by vertices with degree1 (perfect matchings) resp. the 2-core (Hamilton cycles).Next, we establish a couple of properties of tiny and atypical vertices in random graphs.5 emma 2.5. Given δ > , if p ≥ log n/ (3 n ) then G ∼ G n,p a.a.s. satisfies: | ATYP p,δ ( G ) | ≤ n/ (log n ) . Proof.
The left hand side consists of all vertices v ∈ V ( G ) with degree deg( v ) / ∈ (1 ± δ ) np . ByChernoff bounds, we have thatPr[deg( v ) / ∈ (1 ± δ ) np ] ≤ e − Ω δ ( np ) ≤ (log n ) − , with room to spare. Thus, Markov’s inequality implies that the probability that there are atleast n/ (log n ) vertices with degree not in (1 ± δ ) np is at most (log n ) − = o (1), as claimed.The crucial observation in our proof strategy is that tiny and atypical vertices cannot be clumped.In particular, no vertex has too many atypical vertices in its proximity.In the graph process { G i } the property of being tiny or atypical is not monotone. Nevertheless,we expect that these vertices only change slightly over short periods of time. In order to capturethis formally, we make use of the fact that one can generate a subsequence of the graph processas follows: choose p such that G − ∼ G n,p has a.a.s. just a bit less than m edges, and choose p (cid:48) such that G n,p (cid:48) has just a bit more than εm edges. Then the union G + = G − ∪ G n,p (cid:48) containsall graphs from { G i } with m ≤ i ≤ (1 + ε ) m . Moreover, as p (cid:48) (cid:28) p we expect that the tinyand atypical vertices of G − are still scattered, even if we include all the edges from G + . Thefollowing lemma makes this precise. Lemma 2.6.
Let k ≥ be an integer and ε > a constant. There exist positive constants δ ( ε ) and L ( ε, k ) , such that if p ≥ (1 + ε ) log n/ ( kn ) and p (cid:48) ≤ εp then a.a.s. the following holds. Let G − ∼ G n,p and set G + = G − ∪ G n,p (cid:48) and p = 1 − (1 − p )(1 − p (cid:48) ) . Then:(i) for every v ∈ V ( G + ) we have (cid:12)(cid:12) N G + ( v ) ∩ (TINY p ,δ ( G − ) ∪ TINY p ,δ ( G + )) (cid:12)(cid:12) ≤ k − ,(ii) for every v ∈ V ( G + ) we have (cid:12)(cid:12) N G + ( v ) ∩ (ATYP p ,δ ( G − ) ∪ ATYP p ,δ ( G + )) (cid:12)(cid:12) ≤ L ,(iii) for every cycle C ⊆ G + with v ( C ) ≤ k we have (cid:12)(cid:12) V ( C ) ∩ (TINY p ,δ ( G − ) ∪ TINY p ,δ ( G + )) (cid:12)(cid:12) ≤ k − . Proof.
Throughout the proof we assume that V ( K n ) = V ( G + ). We make use of the fact that G + is distributed as G n,p .( i ) We prove a more general statement: if T ⊆ G + is a tree with k ≤ v ( T ) ≤ k vertices, thenit contains at most k − p ,δ ( G − ) ∪ TINY p ,δ ( G + ). Suppose that this is thecase. Let v ∈ V ( G + ) be an arbitrary vertex and assume, towards a contradiction, that there are k vertices u ∈ TINY p ,δ ( G − ) ∪ TINY p ,δ ( G + ) in N G + ( v ). Choose T to be a tree which contains v and a shortest path between v and every such vertex u . Then T has at least 1 + k and atmost 1 + 3 k vertices, and contains k vertices from TINY p ,δ ( G − ) ∪ TINY p ,δ ( G + ), which is acontradiction.Let T ⊆ K n be a tree with k ≤ v ( T ) ≤ k vertices and consider a subset S ⊆ V ( T ) of sizeexactly k . Let E T denote the event that T ⊆ G + and E T,S the event that every vertex in S has at most δnp neighbours in G − which are outside of V ( T ), or at most δnp neighbours in G + outside of V ( T ). Note that E T and E T,S are independent events. Moreover, if there exists atree T ⊆ G + which contains at least k vertices from TINY p ,δ ( G − ) ∪ TINY p ,δ ( G + ) then clearlyboth events E T and E T,S happen for S being a subset of size k of such vertices. We show thatthe probability that E T ∧ E T,S happens for any
T, S is o (1), which implies the statement.First, note that for a fixed tree T ⊆ K n we have Pr[ E T ] = p v ( T ) − . Next, in order for E T,S tohappen, for each vertex in S we need it to have at most δnp edges incident to it in G − , or at6ost δnp edges in G + , with the other endpoint being in V ( G + ) \ V ( T ). The probability thatthis happens is at mostPr[Bin( n − v ( T ) , p ) ≤ δnp ] + Pr[Bin( n − v ( T ) , p ) ≤ δnp ] . Elementary calculations show that this sum can be bounded by e − (1 − ε/ np , for sufficientlysmall δ depending on ε . As the probabilities are independent for different vertices in S , we getPr[ E T,S ] ≤ ( e − (1 − ε/ np ) k . For each k ≤ t ≤ k there are at most (cid:0) nt (cid:1) t t − different trees T ⊆ K n with exactly t vertices,and for each such tree (cid:0) tk (cid:1) choices for S ⊆ V ( T ). Therefore, using union bound over all values of t and all such pairs ( T, S ) we estimate the probability that some E T ∧ E T,S happens as follows:Pr[ (cid:91) ( T,S ) ( E T ∧ E T,S )] ≤ k (cid:88) t = k (cid:18) nt (cid:19)(cid:18) tk (cid:19) t t − · Pr[ E T ∧ E T,S ] ≤ k (cid:88) t = k n t t t · p t − · e − k (1 − ε/ np . As p ≤ (1 + ε ) p and n t p t − · e − k (1 − ε/ np is decreasing in p , this impliesPr[ (cid:91) ( T,S ) ( E T ∧ E T,S )] = O ε,k ( n · (log n ) k · n − − ε/ ) = o (1) . ( ii ) Similarly to the previous case, we prove a statement for trees that implies the desiredresult. Let T ⊆ K n be a tree with L ≤ v ( T ) ≤ L and let S ⊆ V ( T ) be a set of L vertices.Let E T denote the event that T ⊆ G + and E T,S the event that every vertex v ∈ S satisfieseither | N G − ( v ) \ V ( T ) | / ∈ (1 ± δ/ np or | N G + ( v ) \ V ( T ) | / ∈ (1 ± δ/ np . If there exists T ⊆ G + with at least L vertices belonging to ATYP p ,δ ( G − ) ∪ ATYP p ,δ ( G + ), both E T and E T,S happen (where once again S is any subset of L such vertices). For a fixed vertex v ∈ S ,Chernoff bounds show that the probability that v satisfies | N G − ( v ) \ V ( T ) | / ∈ (1 ± δ/ np or | N G + ( v ) \ V ( T ) | / ∈ (1 ± δ/ np is at mostPr[ E T,S ] ≤ Pr[Bin( n − v ( T ) , p ) / ∈ (1 ± δ/ np ] + Pr[Bin( n − v ( T ) , p ) / ∈ (1 ± δ/ np ] ≤ e − γnp , for some γ = γ ( ε ) > δ = δ ( ε ) sufficiently small). Note that these events are independentfor different vertices in S and thereforePr[ E T,S ] ≤ ( e − γnp ) L . Analogous union bound analysis as in part ( i ) yieldsPr[ (cid:91) ( T,S ) ( E T ∧ E T,S )] = O ε,k ( n · (log n ) L · e − Lγnp ) = o (1) , for L large enough depending on ε and k .( iii ) Let C ⊆ K n be a cycle with 3 ≤ v ( C ) ≤ k vertices and consider a subset S ⊆ V ( C ) of sizeexactly k −
1. Let E C and E C,S denote the events as in ( i ) with T replaced by C . Assuming thatthere exists a cycle C ⊆ G + with at least k − p ,δ ( G − ) ∪ TINY p ,δ ( G + ),both events E C and E C,S happen (again, for S being any subset of k − E C,S ] ≤ ( e − (1 − ε/ np ) k − , On the other hand, we have Pr[ E C ] = p v ( C )1 . For each 3 ≤ (cid:96) ≤ k there are less than n (cid:96) cycles C ⊆ K n with exactly (cid:96) vertices, and for each such cycle at most (cid:0) (cid:96)k − (cid:1) many choices for7 ⊆ V ( C ). A union bound over all values of (cid:96) and all possible pairs ( C, S ), shows that theprobability that the property ( iii ) fails is at mostPr[ (cid:91) ( C,S ) ( E C ∧ E C,S )] ≤ k (cid:88) (cid:96) =3 n (cid:96) (cid:18) (cid:96)k − (cid:19) · Pr[ E C ∧ E C,S ] ≤ k (cid:88) (cid:96) =3 n (cid:96) (cid:96) k · p (cid:96) · e − ( k − − ε/ np . We once again obtainPr[ (cid:91) ( C,S ) ( E C ∧ E C,S )] = O ε,k ((log n ) k · n − ( k − ε/ /k ) = o (1) , and the property follows.We note that the proof actually shows that we could replace N G + ( v ) with N (cid:96)G + ( v ) for anyconstant (cid:96) ∈ N . For our purposes, the third neighbourhood suffices. In this subsection we establish some further, more general properties of random graphs whichwe rely on in the later sections. The following is a well known bound on the number of edgesbetween sets of vertices in random graphs (see, e.g. [25, Corollary 2.3]).
Lemma 2.7.
Let p = p ( n ) ≤ . . Then G ∼ G n,p a.a.s. has the property that for every two(not necessarily disjoint) subsets X, Y ⊆ V ( G ) the number of edges with one endpoint in X andthe other in Y satisfies: | e G ( X, Y ) − | X || Y | p | ≤ c (cid:112) | X || Y | np, for some absolute constant c > . Lastly, we show that if p is not too large then the neighbourhood of every two vertices is almostdisjoint. Lemma 2.8. If p = o ( n − / ) then G ∼ G n,p a.a.s. has the property that every two distinctvertices have at most two common neighbours, i.e. | N G ( u ) ∩ N G ( v ) | ≤ for all distinct u, v ∈ V ( G ) .Proof. It is sufficient to show that there are no two vertices with a common neighbourhood ofsize three. The probability that there exists a pair of vertices u, v ∈ V ( G ) violating this propertyis at most (cid:18) n (cid:19)(cid:18) n − (cid:19) p ≤ n p = o (1) . This completes the proof.
Let { G i } denote the random graph process. Recall that Theorem 1.2 states that a.a.s. for every m ≥ ε n log n we have that the subgraph of G m obtained by removing isolated vertices isresilient with respect to containing a perfect matching. As remarked earlier, G m has the samedistribution as G n,m thus an obvious way to prove this statement would be to estimate for each m the probability that G n,m has the resilience property and apply a union bound. Unfortunately,this probability is roughly 1 − e − α · m/n , for some small constant α >
0, which is too weak tocover all the values of m which are of order n log n . This is not surprising as typically random8raphs fail to satisfy a property involving spanning subgraphs with a probability that is onlyexponential in the average degree. We go around this issue by using the following strategy:rather than handling each graph individually, we bundle consecutive graphs in groups of sizeroughly εn log n . The following proposition shows that almost surely all graphs in such a grouphave the desired property. A union bound over constantly many groups thus implies that thetheorem holds for all m ∈ { ε n log n, . . . , Cn log n } , for some large constant C of our choice.The remaining values of m can then be treated one at a time using a result of Sudakov andVu [33], which gives a probability of at least 1 − /n for all m ≥ Cn log n . Proposition 3.1.
For every constant ε > and integer ε n log n ≤ m ≤ n (log n ) , there existpositive constants δ t ( ε ) , δ a ( ε ) , and K ( ε ) such that the random graph process { G i } a.a.s. has thefollowing property: For any integer m ≤ m ≤ (1 + ε/ m the graph obtained by removingall isolated vertices from G m is (1 / − ε, δ t , , δ a , K ) -resilient with respect to having a perfectmatching. We remark that the fact that we use the same ε in the lower bound for m as we do for theresilience is without loss of generality, as we can always choose the smaller of the two for both. Proof overview.
The proof itself is somewhat technical, so let us first give a brief overviewbefore we dive into the details. To handle all graphs G m for m ∈ [ m , (1+ ε/ m ] simultaneouslywe change the way G m is generated (akin to what has been indicated prior to Lemma 2.6).Instead of generating G m starting from an empty graph, we obtain it as follows: sample G − ∼ G n,p and G n,p (cid:48) , and choose a random ordering π of the edges in G n,p (cid:48) . In particular, we choose p and p (cid:48) such that almost surely G − has less than m edges (but not too much less) and G + = G − ∪ G n,p (cid:48) has more than (1 + ε/ m edges (again, not too much more). Now we cangenerate each G m as a union of G − and the first m − e ( G − ) edges according to π ; it is a simpleexercise to show that this is the same as generating G m from ‘scratch’.It is crucial that all the properties of G − and G + that we use (e.g. upper bound on the numberof edges between certain sets in G + , tiny and atypical vertices being far apart in both G − and G + , etc.), are such that they are also satisfied by all graphs ‘squeezed’ in between them. Whilethis is a standard technique in showing hitting-time results, it raises problems in the resiliencesetting. For example, vertices can be tiny or atypical in G m without being tiny or atypical in G − or G + .We circumvent this by defining tiny vertices to be those vertices that are tiny in at least oneof G − or G + with respect to a parameter δ (cid:48) t which is somewhat larger than the value of δ t inthe definition of (1 / − ε, δ t , , δ a , K )-resilience in Proposition 3.1. In this way we can guaranteethat every vertex in G that is tiny with respect to δ t is also tiny in G − or G + with respect to δ (cid:48) t . If we thus allow an adversary to remove all but one incident edge for all vertices that aretiny in G − or G + with respect to δ (cid:48) t , this may result in the removal of more edges than allowedby the definition of (1 / − ε, δ t , , δ a , K )-resilience. Nevertheless, Lemma 2.6 still guaranteesthat all those vertices are sufficiently far apart in G + and thus also in G ⊆ G + . This, in turn,implies that all these vertices can be covered in the matching even though they all have only oneedge left. Atypical vertices are handled similarly by considering atypical vertices in G − or G + with respect to a parameter δ (cid:48) a that is somewhat smaller than the value of δ a in the definitionof (1 / − ε, δ t , , δ a , K )-resilience. The rest of the proof then follows standard arguments forfinding a perfect matching in sparse random graphs (see, e.g. [19, 29]): greedily match all tinyand atypical vertices, in that order, and show that the remaining graph can be equipartitionedin a way that the resulting bipartite graph satisfies Hall’s matching criteria. Proof.
Let p = (1 − ε/ m / (cid:0) n (cid:1) , p (cid:48) = ( ε/ p , and let G + be the union of independent randomgraphs G − ∼ G n,p and G n,p (cid:48) . Then G + is distributed as G n,p , where p = 1 − (1 − p )(1 − p (cid:48) ).9y Lemma 2.7 we have that the number of edges in G − is a.a.s. at most m and the number ofedges in G + is a.a.s. at least (1 + ε/ m .Let δ (cid:48) t = δ . ( ε/ δ (cid:48) a = min { ε/ , δ . ( ε/ } , c = c . , L = max { L . ( ε/ , L . ( ε ) } . With thisat hand, we set the constants given in the statement of the proposition as follows: δ t = δ (cid:48) t / δ a = max { ε/ , δ (cid:48) a } , and K = 2 L .For the rest of the proof consider some m ≤ m ≤ (1 + ε/ m , and let G ⊆ G m be the subgraphobtained by removing all isolated vertices from G m . Let V = V ( G ) denote its vertex set and letTINY and ATYP be sets of vertices defined as:TINY := (TINY p ,δ (cid:48) t ( G − ) ∪ TINY p ,δ (cid:48) t ( G + )) ∩ V, ATYP := (ATYP p ,δ (cid:48) a ( G − ) ∪ ATYP p ,δ (cid:48) a ( G + )) ∩ V. By Lemma 2.5 there are at most n/ log n isolated vertices in G − , thus there are at most thatmany in G m as well. Our choice of constants then implies that for all m ≤ m ≤ (1 + ε/ m we have TINY p,δ t ( G ) ⊆ TINY and ATYP p,δ a ( G ) ⊆ ATYP , where p = m/ (cid:0) v ( G )2 (cid:1) denotes the density of G (note that E ( G ) = E ( G m ) as we only removeisolated vertices). As noted in the discussion before the proof, if we show that for every graph H of the form deg H ( v ) ≤ deg G ( v ) − , if v ∈ TINY , deg G ( v ) − K, if v ∈ ATYP \ TINY , (1 / − ε ) np , otherwise , (1)the graph G − H contains a perfect matching, then this implies that G is (1 / − ε, δ t , , δ a , K )-resilient with respect to having a perfect matching. Note that this follows from the fact thattypical vertices in G have degree at most (1 + δ a ) np and thus removing an (1 / − ε )-fractionof their degree is less than removing (1 / − ε ) np incident edges, since p > (1 + ε/ p . In theremainder of the proof we show that the graph G − H indeed contains a perfect matching.First, note that G satisfies a series of properties: (M1) the maximum degree of G is at most ∆( G ) ≤ log n , (M2) for all X, Y ⊆ V we have e G ( X, Y ) ≤ | X || Y | p + c (cid:112) | X || Y | np , (M3) for all v, u ∈ V we have | N G ( v ) ∩ N G ( u ) | ≤ (M4) for all v ∈ V we have | N G ( v ) ∩ TINY | ≤ | N G ( v ) ∩ ATYP | ≤ L . (M5) | ATYP | ≤ n log n .We show that the properties hold in G + , and hence in any subgraph G ⊆ G + . Indeed, wehave p ≤
10 log n/n (by our assumptions on m ) thus (M1) follows from simple bounds on thebinomially distributed random variable and a union bound over all vertices. Property (M2) isgiven by Lemma 2.7 and (M3) by Lemma 2.8. Next, (M4) holds by our choice of L and byLemma 2.6 applied with k = 2 and ε/ ε . Note that we can indeed apply Lemma 2.6 withthese parameters as p ≥ (1 + ε/
2) log n/ (2 n ). Lastly, (M5) holds by Lemma 2.5.Consider graph H which satisfies (1), and let G (cid:48) = G − H and U := V \ ATYP. If G has anodd number of vertices then we remove a vertex v ∈ TINY which is incident to u / ∈ TINY, andif such a vertex does not exist, then we remove one from V \ TINY. Owing to the property(M4) this does not decrease the degree of any vertex from TINY and decreases the degree of avertex from V \ TINY by at most one. Note that all vertices v ∈ U have degree in G − at least101 − δ (cid:48) a ) np , hence by (M4) we have:deg G (cid:48) ( v, U ) ≥ deg G − ( v ) − deg H ( v ) − deg G ( v, ATYP) − ≥ (1 − δ (cid:48) a ) np − (1 / − ε ) np − L − ≥ (1 / − δ (cid:48) a + ε/ np − L − ≥ (1 / ε/ np , as δ (cid:48) a ≤ ε/
64 and p ≥ p / (1 + ε/ w ∈ ATYP \ TINY. Note that deg G (cid:48) ( w ) ≥ K − − G to achieve aneven number of vertices). By (M4) and our choice of K we have | N G (cid:48) ( w ) ∩ ATYP | ≤ L ≤ K/ K/ w have been matched so far, thus there is an availableone.Let V ( M ) be the set of all vertices saturated in this partial matching. In particular V ( M )contains all vertices of ATYP. Set U := U \ V ( M ) and let G (cid:48)(cid:48) := G (cid:48) [ U ] be the subgraphinduced by the remaining vertices. Observe that by property (M4) the degree of any vertex in U decreases by at most 2 L with respect to its degree in G (cid:48) , hence for all vertices v ∈ U wehave deg G (cid:48)(cid:48) ( v ) ≥ deg G (cid:48) ( v, U ) − deg G (cid:48) ( v, V ( M )) ≥ (1 / ε/ np − L ≥ (1 / ε/ np . Take U = A ∪ B to be a uniformly at random chosen balanced bipartition of the set U . Wenow define the set of all vertices which have significantly less than the expected degree in either A or B as D := { v ∈ U : deg G (cid:48)(cid:48) ( v, A ) < (1 / ε/ | A | p or deg G (cid:48)(cid:48) ( v, B ) < (1 / ε/ | B | p } and call all such vertices degenerate .Firstly, we show that there are not many degenerate vertices. As | A | , | B | ≥ (1 / − o (1)) n by property (M5), we have from Chernoff bounds for hypergeometrically distributed randomvariables that a fixed vertex v is degenerate with probabilityPr[ v is degenerate] ≤ G (cid:48)(cid:48) ( v, A ) < (1 / ε/ | A | p ] ≤ e − γnp ≤ n − γ , for some γ = γ ( ε ) >
0. Consequently, by Markov’s inequality there are at most n − γ degeneratevertices, i.e. | D | ≤ n − γ . Next we show that the degenerate vertices cannot be ‘too close’ in G (cid:48)(cid:48) . Claim 3.2.
There exists a positive constant L ( ε ) such that a.a.s. for every v ∈ U we have | N G (cid:48)(cid:48) ( v ) ∩ D | < L .Proof. Consider some vertex v ∈ V ( G (cid:48)(cid:48) ) and a subset D v ⊆ N G (cid:48)(cid:48) ( v ) of size L . What is theprobability that a random equipartition A ∪ B of V ( G (cid:48)(cid:48) ) makes all the vertices in D v degenerate?First, by property (M3) we know that for every vertex u ∈ D v there are at least(1 / ε/ np − L ≥ (1 / ε/ np (2)vertices in its neighbourhood in G (cid:48)(cid:48) which do not belong to the neighbourhood of any othervertex in D v \ { u } . Let us denote such vertices with N ∗ u .If all vertices in D v are degenerate, then there has to exist a subset D (cid:48) v ⊆ D v of size at least L/ D (cid:48) v either have too few neighbours into A or too few neighbours into B .11y symmetry we may assume that this set is A . That is, we have | A ∩ N ∗ u | ≤ (1 / ε/ | A | p for every u ∈ D (cid:48) v . Therefore, (cid:12)(cid:12)(cid:12) A ∩ (cid:91) u ∈ D (cid:48) v N ∗ u (cid:12)(cid:12)(cid:12) ≤ (1 / ε/ | A | p | D (cid:48) v | . Let E D (cid:48) v denote the event of this happening. From (2) we have (cid:12)(cid:12)(cid:12) (cid:91) u ∈ D (cid:48) v N ∗ u (cid:12)(cid:12)(cid:12) ≥ (1 / ε/ np | D (cid:48) v | . Thus, as A is a randomly chosen subset of linear size, Chernoff bounds for hypergeometricallydistributed random variables show Pr[ E D (cid:48) v ] ≤ e − γLnp , (3)for some γ = γ ( ε ) > L . To summarise, we can bound the probability that allthe vertices in D v are degenerate by applying (3) together with a union bound over all possiblechoices for D (cid:48) v . Finally, we take a union bound over all vertices v and sets D v . There are n choices for v and at most (cid:0) (log n ) L (cid:1) choices for D v ⊆ N G (cid:48)(cid:48) ( v ) of size L (which follows from (M1)).The expected number of vertices for which there exists a set D v of size L consisting solely ofdegenerate vertices is then at most n · (log n ) L · L · e − γLnp = o (1) , for L > L large enough no such set exists with probability1 − o (1), as claimed.Similarly as before, we greedily construct a partial matching M D that saturates all the degeneratevertices. For an arbitrary degenerate vertex v ∈ D we have deg G (cid:48)(cid:48) ( v ) ≥ (1 / ε/ np and asthere cannot be more than L degenerate vertices in N G (cid:48)(cid:48) ( v ) by Claim 3.2, there is a vertexavailable to match v to.Let V ( M D ) be the set of all vertices saturated in this partial matching and let A (cid:48) := A \ V ( M D )and B (cid:48) := B \ V ( M D ). Again by Claim 3.2, for all v ∈ ( A ∪ B ) \ V ( M D ) we getdeg G (cid:48)(cid:48) ( v, A (cid:48) ) ≥ (1 / ε/ | A | p − L ≥ (1 / ε/ np , and analogously deg G (cid:48)(cid:48) ( v, B (cid:48) ) ≥ (1 / ε/ | B | p − L ≥ (1 / ε/ np , as | A | , | B | > (1 / − o (1)) n . However, we might not have | A (cid:48) | = | B (cid:48) | any more. Assume w.l.o.g.that | A (cid:48) | > | B (cid:48) | and note that | A (cid:48) | ≤ | B (cid:48) | + | V ( M D ) | . In order to find a balanced bipartitionwe thus need to redistribute at most | D | ≤ n − γ vertices, for some γ >
0. To achieve this webuild a 2- independent set in A (cid:48) , i.e. an independent set in which no two vertices have a commonneighbour, of size at least n − γ . Recall, from property (M1) we have ∆( G ) ≤ log n , thus astraightforward greedy construction shows that there exists a 2-independent set of size at least n/ log n ≥ n − γ in A (cid:48) , which is more than enough for our purposes.Let A (cid:48)(cid:48) and B (cid:48)(cid:48) be the two sets after moving the vertices belonging to the 2-independent set into B (cid:48) . Then | A (cid:48)(cid:48) | = | B (cid:48)(cid:48) | (by construction) and vertices in A (cid:48)(cid:48) (respectively, B (cid:48)(cid:48) ) have degree at least(1 / ε/ np − ≥ (1 / ε/ np in B (cid:48)(cid:48) (respectively, A (cid:48)(cid:48) ), as we moved a 2-independentset.It remains to find a perfect matching in the bipartite graph G (cid:48)(cid:48) [ A (cid:48)(cid:48) , B (cid:48)(cid:48) ]. This is easily achievedthrough Hall’s matching criteria (see, e.g. [10]). Recall its statement: if each subset S ⊆ A (cid:48)(cid:48) of12ize at most | A (cid:48)(cid:48) | / | S | neighbours in B (cid:48)(cid:48) (and similarly for S ⊆ B (cid:48)(cid:48) ), then G (cid:48)(cid:48) [ A (cid:48)(cid:48) , B (cid:48)(cid:48) ]contains a perfect matching. We now verify that this is indeed the case.Let S ⊆ A (cid:48)(cid:48) be an arbitrary subset of size s . The number of edges between any two disjointsubsets of G (cid:48)(cid:48) of size s ≤ n/ s ( np / c √ np ), by property (M2). On the otherhand however, the minimum degree condition in G (cid:48)(cid:48) yields that the number of edges between S and B (cid:48)(cid:48) is at least s (1 / ε/ np , which implies | N ( S ) | > s . This completes the proof of theproposition.Having Proposition 3.1 at hand we now complete the proof. Let us first, as promised in theintroduction, restate Theorem 1.2 in terms of stronger resilience. Theorem 3.3.
Let ε > be a constant and consider the random graph process { G i } . Thereexist positive constants δ t ( ε ) , δ a ( ε ) , and K ( ε ) such that a.a.s. for every m ≥ ε n log n we havethat the graph obtained from G m by deleting all isolated vertices is (1 / − ε, δ t , , δ a , K ) -resilientwith respect to containing a perfect matching.Proof. Let C be a sufficiently large constant. If m ≥ Cn log n then the random graph G n,m doesnot contain atypical vertices. Therefore, by [33, Theorem 3.1] the statement holds for everysuch (fixed) m with probability at least 1 − e − α · C log n , for some small constant α >
0, thus aunion bound implies that it holds for all m ≥ Cn log n simultaneously. As for the rest, considerintervals of the form (cid:20) iε n log n, i + 1) ε n log n (cid:19) for i ∈ { , . . . , C ε } , where C ε is such that the last interval contains Cn log n . For each intervalthe conclusion of Proposition 3.1 holds with probability 1 − o (1), thus a union bound shows thatit a.a.s. holds for all intervals. This concludes the proof. Proof of Theorem 1.2.
The assertion follows directly from Lemma 2.4 and Theorem 3.3.
The so-called P´osa’s rotation-extension method introduced in [31] is nowadays a standard ap-proach for constructing Hamilton cycles in random graphs, cf. also [4, 14, 15, 22, 24]. Theproblem with having edge probabilities near (or even below) the connectivity threshold, is thatthen the random graph does not satisfy the expansion properties needed to apply the method.We go around this by partitioning the vertex set into typical and atypical vertices. The subgraphinduced by the vertices of typical degree satisfies the expansion properties required to apply therotation-extension technique and we thus get a Hamilton cycle in this subgraph by a standardapproach. Our new contribution is to show how to extend this cycle to also contain all atypicalvertices.Towards this goal we make use of the facts given by Lemma 2.6, that is that atypical verticesdo not clump. This allows us to modify the rotation-extension procedure from [27] such that itonly uses the expansion properties of typical vertices. In the next section we review some basicnotions and state necessary lemmas which are then used in the subsequent section to deriveTheorem 1.4.
The central notion of the rotation-extension method is that of boosters . A booster is a non-edgein a graph G whose existence would increase the length of a longest path in G or close a Hamilton13ath to a Hamilton cycle. The idea behind the rotation-expansion technique is that a graphwhich is not Hamiltonian contains so many boosters that adding a few random edges is highlylikely to contain one of them. The name ‘rotation-extension’ comes from the way how boostersare obtained ( rotation ) and the fact that a longest path of every non-Hamiltonian graph can beincreased by adding an edge in place of a booster ( extension ). We now make this precise. Definition 4.1 (Boosters) . Given a graph Γ, we say that a non-edge { u, v } / ∈ E (Γ) is a booster with respect to Γ, if either Γ + { u, v } is Hamiltonian or adding { u, v } to Γ increases the lengthof a longest path. For a vertex v ∈ V (Γ), we denote by B Γ ( v ) the set of boosters associatedwith v : B Γ ( v ) = { u ∈ V (Γ) \ ( N Γ ( v ) ∪ { v } ) : { v, u } is a booster } . A standard strategy for implementing the rotation-extension technique is to split the givengraph into two graphs: a ‘backbone’ graph responsible for obtaining boosters, and the remainderresponsible for finding real edges corresponding to boosters. As we are dealing with subgraphsof random graphs, rather than random graphs themselves, it is convenient to capture the mainpseudorandom properties which are used.
Definition 4.2 (Backbone graph) . Given α, q ∈ (0 ,
1) and an integer K ≥
1, we say that agraph Γ with n vertices is an ( α, K, q )- backbone graph if there exists a partition of its vertex set V (Γ) = U ∪ W ∪ W such that the following holds: (P1) | W ∪ W | ≤ n log n , (P2) for every v ∈ W we have deg Γ ( v ) = 2 and for every u ∈ W we have deg Γ ( u ) ≥ K , (P3) for every v ∈ W we have N ( v ) ∩ W = ∅ , (P4) for every v ∈ V (Γ) we have | N ( v ) ∩ W | ≤ | N ( v ) ∩ W | ≤ K , and (P5) for all S ⊆ U we have | N Γ ( S ) | ≥ (cid:40) | S |√ nq, if | S | < K/q, (1 / α/ n, if | S | ≥ K/q.
The role of the sets W and W and properties (P1)–(P4) is to capture properties of tiny andatypical vertices in a random graph with density q . Property (P5) states that the subgraphinduced by typical vertices has good expansion properties.The next lemma shows that a backbone graph contains many boosters. It can be proven similarlyas [27, Lemma 3.2], with slight modifications which allow us to deal with the vertices in W and W . We defer the proof to Section 4.3. Lemma 4.3.
For every α, δ > , there exists a positive constant K ( α ) such that the followingholds for q ≥ δ log n/n . Let G q be a graph with n vertices satisfying the property of Lemma 2.7for q (as p ) and some constant c (cid:48) (as c ), and H q ⊆ G q a graph with ∆( H q [ U ]) ≤ (1 / − α ) nq ,for some U ⊆ V ( G q ) . Then an ( α, K, q ) -backbone graph Γ with a witness partition V (Γ) = U ∪ W ∪ W , such that Γ[ U ] = G q [ U ] − H q [ U ] , is either Hamiltonian or there are at least (1 / α ) n vertices v ∈ U such that | B Γ ( v ) ∩ U | ≥ (1 / α ) n . Note that rather than asking for Γ = G q − H q , we only require the subgraph of Γ induced by U to be given by G q [ U ] − H q [ U ]. The way we exploit such a weaker requirement becomes apparentin the next section. As the notation q in the previous lemma already indicates, we obtain an( α, K, q )-backbone graph Γ by considering a random subgraph with density q of some graph G .The next lemma states that given a sufficiently sparse graph Γ with many boosters, a randomgraph with appropriate density is likely to contain plenty of edges corresponding to them.14 emma 4.4. For every α > there exists a positive constant µ ( α ) such that for p ≥ log n/ (3 n ) ,the random graph G ∼ G n,p a.a.s. satisfies the following. Let Γ be a graph with e (Γ) ≤ µn p and U ⊆ V ( G ) a subset of vertices such that Γ[ U ] ⊆ G . If there are at least (1 / α ) n vertices v ∈ U such that | B Γ ( v ) ∩ U | ≥ (1 / α ) n , then there exists a vertex v ∈ U satisfying | N G ( v, B Γ ( v ) ∩ U ) | > np/ . The proof of the lemma is fairly standard and goes along the lines of the proof of [27, Lemma3.5]; we include it for completeness in Section 4.3.
Similarly as in the case of perfect matchings, instead of proving Theorem 1.4 directly, we firstshow a proposition which considers only a small range of m , until m = Cn log n . Proposition 4.5.
For every constant ε > and integer m ≥ (1 + ε ) n log n there exist positiveconstants δ t ( ε ) , δ a ( ε ) , and K ( ε ) such that the random graph process { G i } a.a.s. has the followingproperty: For every integer m ≤ m ≤ (1 + ε/ m , the -core of G m is (1 / − ε, δ t , , δ a , K ) -resilient with respect to having a Hamilton cycle. The general strategy of the proof is similar as in the case of perfect matchings. We first sampletwo random graphs G − ∼ G n,p and G n,p (cid:48) and look at their union G + = G − ∪ G n,p (cid:48) . The choiceof densities p and p (cid:48) is such that for all values of m ∈ [ m , (1 + ε/ m ] the graph G m is a.a.s.‘in between’, that is G − ⊆ G m ⊆ G + .Let G denote the 2-core of G m and let H be a graph removed by the adversary, as in Defini-tion 2.3. We split the graph G − H into a sparse graph Γ (cid:48) which mostly contains edges from G − − H (edges outside of G − are borrowed to handle atypical vertices), and the rest G − H − Γ (cid:48) .Ideally, we would like that Γ (cid:48) is a backbone graph. For m close to n log n/
6, the graph G containsshort paths between two tiny vertices, some of which exist in Γ (cid:48) as well preventing it from satis-fying (P3). We circumvent this by constructing a new graph Γ from Γ (cid:48) by contracting all pathsof length at most two between tiny vertices. As tiny vertices do not clump, this has only a mildimpact on the structure of the graph and all the other properties remain satisfied. Due to thesecontractions the graph Γ is not a subgraph of G − H any more, however, the subgraph inducedby typical vertices which are not a part of contracted paths, is. This allows us to apply Lemma4.3 to conclude that such Γ is either Hamiltonian in which case we are done by ‘unrolling’ thecontracted vertices, or contains many boosters between ‘real’ vertices of G . Using Lemma 4.4,we subsequently show that many such boosters appear in G − H which allows us to complete aHamilton cycle.We point out that by requiring m just a bit larger, that is m ≥ (1 + ε ) n log n/
4, would be enoughfor (P3) to hold already in Γ (cid:48) and the contraction process would not be necessary.
Proof.
Let p = (1 − ε/ m / (cid:0) n (cid:1) , p (cid:48) = ( ε/ p , and let G + be the union of independent randomgraphs G − ∼ G n,p and G n,p (cid:48) . Then G + is distributed as G n,p , where p = 1 − (1 − p )(1 − p (cid:48) ).By Lemma 2.7 we a.a.s. have both e ( G − ) ≤ m and e ( G + ) ≥ (1 + ε/ m .Let δ (cid:48) t = δ . ( ε/ δ (cid:48) a = min { ε/ , δ . ( ε/ } , c = c . , L = max { L . ( ε/ , L . ( ε ) } , µ = µ . ( ε/ c (cid:48) = c (cid:48) ( c, µ ) sufficiently large (cf. (4)), and set the constants given in the statement ofthe proposition as follows: δ t = δ (cid:48) t / , δ a = max { ε/ , δ (cid:48) a } , and K = max { L, (cid:0) c (cid:48) /ε (cid:1) } . For the rest of the proof consider some m ≤ m ≤ (1 + ε/ m . Let G be the 2-core of G m and set V = V ( G ). Note that G can be obtained using the following procedure: initially set15 = G m , and as long as G contains a vertex of degree at most one, remove it. Let R denotethe set of removed vertices. The definition of the procedure then implies that there are at most | R | edges incident to R in G m . By Lemma 2.6 applied to G + (with k = 3), each vertex has atmost two neighbours in TINY p ,δ (cid:48) t ( G − ) thus no vertex which is not in TINY p ,δ (cid:48) t ( G − ) can everbe removed. This also implies that the degree of every vertex decreases by at most 2. As thus R ⊆ TINY p ,δ (cid:48) t ( G − ), Lemma 2.5 implies | R | = O ( n/ log n ).Similarly as in the proof of Proposition 3.1, we define TINY and ATYP as follows:TINY := (TINY p ,δ (cid:48) t ( G − ) ∪ TINY p ,δ (cid:48) t ( G + )) ∩ V, ATYP := (ATYP p ,δ (cid:48) a ( G − ) ∪ ATYP p ,δ (cid:48) a ( G + )) ∩ V. Using the previous observation on the number of removed vertices and edges, it is easy to seethat for all m ≤ m ≤ (1 + ε/ m we haveTINY p,δ t ( G ) ⊆ TINY and ATYP p,δ a ( G ) ⊆ ATYP , where p = e ( G ) / (cid:0) v ( G )2 (cid:1) . Consider a graph H ⊆ G such thatdeg H ( v ) ≤ deg G ( v ) − , if v ∈ TINY , deg G ( v ) − K, if v ∈ ATYP \ TINY , (1 / − ε ) np , otherwise . We show that G − H contains a Hamilton cycle, which in turn implies that G is (1 / − ε, δ t , , δ a , K )-resilient with respect to Hamiltonicity (as in the case of perfect matchings).The key to our proof is the fact that tiny and atypical vertices cannot be clumped up in G ,captured by the following properties: (H1) for all v ∈ V we have | N G ( v ) ∩ TINY | ≤ | N G ( v ) ∩ ATYP | ≤ L , (H2) every cycle C ⊆ G of size at most 6 contains at most one vertex from TINY.By Lemma 2.6 applied for k = 3 and ε/ ε , one easily checks that properties (H1) and (H2)hold in G + , and hence in any subgraph G ⊆ G + . Note that we can indeed apply Lemma 2.6with such parameter, as p ≥ (1 + ε/
2) log n/ (3 n ).In the first step of the proof we carefully construct a backbone graph Γ. Set q = ( µ/ p , where µ = µ . ( ε/ G q obtained by keeping every edge of G − independently withprobability µ/
4. Moreover, remove from G q all vertices which are not in V . By the discussionfrom the beginning of the proof, removing such vertices decreases the degree of every vertex in G q by at most 2. We now show that G q a.a.s. has certain properties.As G − a.a.s. satisfies the assertion of Lemma 2.7, we also have that G q a.a.s. satisfies | e G q ( X, Y ) − | X || Y | q | ≤ c (cid:48) (cid:112) | X || Y | nq, (4)for every two subsets X, Y ⊆ V , for some sufficiently large constant c (cid:48) depending on c and µ .Next, we claim that most of the vertices in V \ ATYP have degree at least (1 − δ (cid:48) a ) nq in G q , aswell as degree at most (1 / − ε/ nq in H q = G q ∩ H . Fix v ∈ V \ ATYP. By Chernoff boundswe get Pr[deg G q ( v ) < (1 − δ (cid:48) a ) nq ] + Pr[deg H q ( v ) > (1 / − ε/ nq ] ≤ e − Ω ε ( nq ) ≤ n − γ , for some γ = γ ( ε ) >
0. Therefore, by Markov’s inequality there are at most n − γ vertices whichsatisfy at least one of these two conditions. We call all such vertices degenerate and denote themby D . Similarly as atypical vertices, such vertices cannot be clumped in G , and thus neither in G q . This is formalised in the following claim. 16 laim 4.6. There exists a positive constant L ( ε ) such that a.a.s. for every v ∈ V we have | N G ( v ) ∩ D | < L .Proof. The proof goes along the lines of that of Claim 3.2. We omit the details.In conclusion, G q a.a.s. satisfies (4), every vertex in V \ (ATYP ∪ D ) has degree at least (1 − δ (cid:48) a ) nq in G q and at most (1 / − ε/ nq in H q , and no vertex has more than L vertices from D in itssecond neighbourhood. From now on we choose one such graph G q . In particular, (4) implies e ( G q ) ≤ ( µ/ n p .We now construct a graph Γ (cid:48) which is ‘almost’ a backbone graph, and then convert it into atrue backbone graph, thereby overcoming the issue of two tiny vertices being on a short path, asdiscussed previously. Let W (cid:48) := TINY, W (cid:48) := (ATYP ∪ D ) \ TINY, and U (cid:48) := V \ ( W (cid:48) ∪ W (cid:48) ).Take Γ (cid:48) to be a graph on the vertex set V obtained by taking all edges in G q [ U (cid:48) ] − H q [ U (cid:48) ] andadding some of the edges of G − H incident to vertices in W (cid:48) ∪ W (cid:48) such that for every v ∈ W (cid:48) we have deg Γ (cid:48) ( v ) = 2, and for every u ∈ W we have deg Γ (cid:48) ( u ) ≥ K − v ∈ W have deg Γ (cid:48) ( v ) = 2 we potentially remove at most two incident edges to a vertex u ∈ W (cid:48) ).Note that e (Γ (cid:48) ) ≤ ( µ/ n p . Recall that property (P3) does not necessarily hold in Γ (cid:48) , thuswe cannot claim it is a backbone graph. We take care of this issue as follows: let Γ be a graphobtained from Γ (cid:48) by contracting every uv -path of length at most two, where u, v ∈ W (cid:48) , andkeeping exactly two edges incident to the newly obtained vertex, namely the ones incident to u resp. v in Γ (cid:48) that are not a part of the uv -path. We also remove all multi-edges and loops.Observe that Γ is well-defined since there cannot exist a vertex w ∈ V which belongs to twosuch paths uv -paths, due to property (H1).In order to show that Γ is a backbone graph, we first define a witness partition U ∪ W ∪ W .Let X be the set of vertices of Γ obtained by contractions, Y be the set of all vertices in W (cid:48) ∪ U (cid:48) that are inner vertices of some contracted path, and Z the set of all vertices of W (cid:48) that are theendpoints of such paths. We now define W := ( W (cid:48) \ Z ) ∪ X, W := W (cid:48) \ Y, and U := U (cid:48) \ Y. In other words, W consists of all newly formed vertices (obtained by contractions) as well asall remaining vertices of W (cid:48) , and W and U consist of all vertices in W (cid:48) resp. U (cid:48) that are not apart of any contracted path.We are now ready to show that Γ is an ( ε/ , K/ , q )-backbone graph. Recall that | V | ≥ n − n/ log n . For every vertex v ∈ U ∪ W we havedeg Γ ( v ) ≥ deg Γ (cid:48) ( v ) − , (5)by using property (H1). We now check all the requirements of Definition 4.2:(P1) follows by Lemma 2.5 and an upper bound on the size of D : | W ∪ W | ≤ | ATYP | + | D | ≤ n log n + n − γ ≤ n n . (P2) follows by construction. Indeed, if for any v ∈ W we have deg Γ ( v ) <
2, then one easilychecks that we arrive to a contradiction with either (H1) or (H2). The second part of theproperty follows from (5).(P3) also holds by construction: if there still exists a path of length at most two between twovertices from W , then this would contradict (H1).(P4) follows from (H1), Claim 4.6, and our choice of K , since all the edges incident to thevertices in W already exist in Γ (cid:48) and as the number of vertices from W that are in N G ( v ) ofany vertex v can at most double in Γ. 17astly, we show the expansion properties of vertices in U , i.e. property (P5). By (5) and thefact that N Γ (cid:48) ( S ) ⊆ N Γ (cid:48) ( S (cid:48) ) whenever S ⊆ S (cid:48) , it suffices to show | N Γ (cid:48) ( S ) | ≥ (cid:40) | S |√ nq + | S | , if | S | < K/ (10 q ) , (1 / ε/ n + | S | , if | S | = (cid:98) K/ (10 q ) (cid:99) . For every v ∈ U we havedeg Γ (cid:48) ( v, U ) = deg G q ( v ) − deg H q ( v ) − deg G q ( v, W ∪ W ) ≥ (1 − δ (cid:48) a ) nq − (1 / − ε/ nq − K/ − ≥ (1 / ε/ nq, by (H1), Claim 4.6, and our choice of K and δ (cid:48) a . Take S ⊆ U to be an arbitrary subset of size | S | ≤ K/ (10 q ) and let T := N Γ (cid:48) ( S ) ∩ U . From the previously obtained bound on deg Γ (cid:48) ( v, U ),we have e Γ (cid:48) ( S, T ) ≥ | S | (1 / ε/ nq. Assume towards a contradiction that | T | < | S |√ nq . Then, from Γ (cid:48) [ U ] ⊆ G q [ U ] and (4) wederive | S | (1 / ε/ nq ≤ e Γ (cid:48) ( S, T ) ≤ | S || T | q + c (cid:48) (cid:112) | S || T | nq ≤ | S | ( nq ) / q + 2 c (cid:48) | S | ( nq ) / , which is a contradiction. On the other hand, if | S | = (cid:98) K/ (10 q ) (cid:99) then, assuming | T | < (1 / ε/ n + εn/
16, again from (4) we have | S | (1 / ε/ nq ≤ e Γ (cid:48) ( S, T ) ≤ | S || T | q + c (cid:48) (cid:112) | S || T | nq < | S | (1 / ε/ nq + ε | S | nq/ , where the last inequality follows from our choice of K . We have a contradiction once again.To conclude, we obtained an ( ε/ , K/ , q )-backbone graph Γ with the witness partition V (Γ) = U ∪ W ∪ W and at most ( µ/ n p edges, and graphs G q and H q applicable by Lemma 4.3.Note that, by the construction of Γ, the following is now true: if for any subset of edges E (cid:48) ⊆ E ( G − [ U ] − H ) the graph Γ + E (cid:48) is Hamiltonian, then G − H is Hamiltonian as well.Indeed, let x uv ∈ X be some vertex obtained by contracting a uv -path in Γ (cid:48) . The two edgesincident to x uv in Γ + E (cid:48) necessarily lie on a Hamilton cycle. Moreover, as they correspond totwo edges in Γ (cid:48) ⊆ G − H , one incident to u and the other to v , by splitting every such vertex x uv back into the uv -path we obtain a Hamilton cycle in G − H .The following claim, akin to [4, Lemma 3.4], allows us to complete the proof. Claim 4.7. [4]
If for every subset E (cid:48) ⊆ E ( G − [ U ] − H ) of | E (cid:48) | ≤ n edges such that Γ + E (cid:48) is notHamiltonian there is a vertex v ∈ U satisfying | N G ( v ) ∩ B Γ+ E (cid:48) ( v ) | > deg H ( v ) , then G − H is Hamiltonian. Let E (cid:48) ⊆ E ( G − [ U ] − H ) be a set of edges of size at most | E (cid:48) | ≤ n . It is not too difficult tosee that Γ + E (cid:48) is an ( ε/ , K/ , q )-backbone graph. Indeed, none of the properties (P1)–(P3)can be violated by adding edges with both endpoints in U . Similarly, the expansion property(P5) is not affected by addition of edges. Lastly, property (P4) holds as Γ + E (cid:48) ⊆ G − [ U ] − H and by referring to (H1). By Lemma 4.3 applied with G q (for ε/ α ) we get that the setof vertices v ∈ U such that | B Γ+ E (cid:48) ( v ) | ≥ (1 / ε/ n is of size at least (1 / ε/ n . As e (Γ + E (cid:48) ) ≤ µn p , by Lemma 4.4 applied with ε/ α and G − as G , there exists a vertex v ∈ U with | N G − ( v ) ∩ B Γ+ E (cid:48) ( v ) | > np / > deg H ( v ). Finally, Claim 4.7 implies that the graph G − H is Hamiltonian. 18aving Proposition 4.5, the proof of the following theorem and Theorem 1.4 are identical to theproofs of Theorem 3.3 and Theorem 1.2, using [27, Theorem 1.1] instead of [33, Theorem 3.1]to handle m ≥ Cn log n . Theorem 4.8.
Let ε > be a constant and consider the random graph process { G i } . Thereexist positive constants δ t ( ε ) , δ a ( ε ) , and K ( ε ) such that a.a.s. for every m ≥ ε n log n we havethat the -core of G m is (1 / − ε, δ t , , δ a , K ) -resilient with respect to being Hamiltonian. Let us first give a brief outline of how to apply P´osa’s rotation-extension technique to backbonegraphs with the goal of constructing long paths.Assume Γ is a backbone graph and let P = v , . . . , v (cid:96) be a path in Γ. If { v , v (cid:96) } is an edge inΓ then such a path can be closed into a cycle. If the obtained cycle does not cover all vertices,then one easily checks that properties (P2)–(P5) imply connectivity of Γ and we can thus extend P into a path longer than P .Suppose that P cannot be extended and let { v , v i } be an edge in Γ for some 2 ≤ i ≤ (cid:96) −
1. Thenthe path P (cid:48) = v i − , . . . , v , v i , . . . , v (cid:96) is another path in Γ of the same length (cid:96) . We say that P (cid:48) is obtained from P by a rotation around the endpoint v , with pivot point v i , and broken edge { v i − , v i } (see, Figure 1). Observe that by performing a rotation we can now possibly obtaina cycle by adding the edge { v i − , v (cid:96) } as well. Otherwise, we perform more rotations to obtainmore boosters. The rotation is repeated until we find a closing edge in Γ. v v v i − v i v i +1 v (cid:96) − v (cid:96) P v v v i − v i v i +1 v (cid:96) − v (cid:96) P (cid:48) Figure 1: Rotation of the the path P around the fixed endpoint v , with pivot point v i , and the brokenedge { v i − , v i } . Pairs of red vertices denote the endpoints of the paths, i.e. boosters. In our setting, we have to be careful with vertices of low degree. This is why we need properties(P2)–(P4). We illustrate in particular why we need property (P3). Suppose P = v , . . . , v (cid:96) is a longest path in Γ such that v , v i − ∈ W for some v i which is the only neighbour (otherthan v ) of v on the path P . Rotating around v with pivot point v i yields a path P (cid:48) = v i − , . . . , v , v i , . . . , v (cid:96) . Now, the only rotation that can be performed around v i − brings thepath P (cid:48) back to where we started from. In conclusion, in such a situation we cannot prove thatΓ contains many boosters. Property (P3) guarantees that this cannot occur. Proof of Lemma 4.3.
Let P = v , . . . , v (cid:96) be a longest path in Γ. For a subset of vertices Z ⊆ V ( P ) we write Z + := { v i +1 : v i ∈ Z } and Z − := { v i − : v i ∈ Z } . For a vertex z we abbreviate { z } + to z + and { z } − to z − .We first show that there exists a longest path with both endpoints in U . Suppose v ∈ W .Then by (P2) and (P4) we know that there are at least K neighbours u ∈ N Γ ( v ) on the path P which have u − ∈ U . Thus, by performing a single rotation around v with pivot point u we get apath P (cid:48) with an endpoint in U . Similarly, if v ∈ W its only other neighbour on the path u has u − belonging to either U or W , by (P3). In any case, by performing at most two rotations wereach a path P (cid:48) with an endpoint in U . Repeating the same argument for v (cid:96) yields the claimedpath P ∗ with both endpoints in U . Phase 1: Initial rotations
The first phase consists of repeatedly rotating the first endpoint as in [27, Lemma 3.2
Step 1 ]in order to obtain a set of at least n/ (cid:96) that all contain the19ame vertices and all end in v (cid:96) . Compared to [27], the only difference in our argument is thatwe ignore all pivot points u with at least one of u , u + , and u − not belonging to U . As everyvertex can have at most K such neighbours u , one easily sees that all the calculations of [27]essentially remain the same. Indeed, starting with X = { v } and denoting by X i the set ofendpoints obtained by exactly i rotations, we get | X i +1 | ≥ (cid:16) | N Γ ( X i ) | − K | X i | − i (cid:88) j =0 | X i | (cid:17) . Using identical calculations as in [27] yields | X i +1 | ≥ ( nq/ ( i +1) / , for all i ≥ | X i | ≤ K/q .Once we reach a set | X m | = max { , (cid:98) K/q (cid:99)} , which is easily observed to happen after at most O (log n/ log log n ) many steps, the above bound on the size of X i +1 , together with property(P5), immediately implies | X m +1 | ≥ n/ Phase 2: Terminal rotation
At the end of the first phase we have a set X of at least n/ Y denotethe set of endpoints that can be generated by exactly one more rotation starting from X .In [27, Lemma 3.2 Step 2 ] it is shown that by partitioning the path P into appropriately manyintervals, denoted by P i , one can define pairs of vertex sets ( X i , Y i ) such that there is no edgebetween X i and Y i in G q − H q , for every i . From Lemma 2.7 we know lower bounds on thenumber of edges in E G q ( X i , Y i ) and thus all these need to belong to H q . In [27, Lemma 3.2 Step2 ] it is shown that | Y | < (1 / α ) n implies a contradiction to the degree assumption of H q .The only difference in our case is that we again need to ignore all pivot points v + ∈ P i with v / ∈ U (which should belong to Y i now), and similarly v − . However, as every interval P i (in line4 of the proof of Step 2 ) is of size at least n (log log n ) / / (4 log n ) and by (P1) each interval P i has at most o ( | P i | ) such ‘bad’ vertices, we can simply ignore them since this contributes o ( n q )many edges. All remaining calculations from [27, Lemma 3.2] remain the same and we obtain aset of new endpoints of size at least (1 / α ) n . Phase 3: Rotating the other endpoint
Exactly as in [27, Lemma 3.2
Step 3 ] for every of the newly obtained (1 / α ) n endpoints, weanalogously rotate the other endpoint to obtain the intended result.We wrap-up by giving a proof of Lemma 4.4. Proof of Lemma 4.4.
Consider a graph Γ with at most µn p edges and a subset U ⊆ V ( G ) suchthat Γ[ U ] ⊆ G and A := { v ∈ U : | B Γ ( v ) ∩ U | ≥ (1 / α ) n } is of size | A | ≥ (1 / α ) n . Take A (cid:48) ⊆ A to be a set of size | A (cid:48) | = αn/
2, and for every v ∈ A (cid:48) let B (cid:48) ( v ) := ( B Γ (cid:48) ( v ) ∩ U ) \ A (cid:48) and note that | B (cid:48) ( v ) | ≥ (1 / α/ n . By Chernoff bounds wehave that the probability of a fixed vertex v ∈ A (cid:48) having fewer than np/ B (cid:48) ( v ) in the graph G is at most e − Ω α ( np ) . Since A (cid:48) is disjoint from all the sets B (cid:48) ( v ),these events are independent for different vertices v, u ∈ A (cid:48) . Therefore, the probability that novertex v ∈ A (cid:48) has more than np/ B (cid:48) ( v ) in G is bounded by e − Ω α ( n p ) . Hence, the probability of failure is upper bounded byPr ≤ e − Ω α ( n p ) · (cid:88) e (Γ (cid:48) ) ≤ µn p Pr[Γ (cid:48) ⊆ G ] . The probability for a fixed graph with t edges to be a subgraph of G is p t and thus by a union20ound over all such graphs, we getPr ≤ e − Ω α ( n p ) · µn p (cid:88) t =1 (cid:18)(cid:0) n (cid:1) t (cid:19) p t ≤ e − Ω α ( n p ) µn p (cid:88) t =1 (cid:18) en pt (cid:19) t . One easily sees that for 0 < µ ≤ ≤ t ≤ µn p and hencewe may substitute t = µn p to concludePr ≤ e − Ω α ( n p ) · ( µn p ) · (cid:18) eµ (cid:19) µn p = e − Ω α ( n p ) · e O ( µ log (1 /µ ) n p ) = o (1) , for sufficiently small µ depending on α . Acknowledgements.
The third author would like to thank Michael Krivelevich for directinghis attention to [22] which helped in making some of the arguments cleaner.
References [1] M. Ajtai, J. Koml´os, and E. Szemer´edi. First occurrence of Hamilton cycles in randomgraphs.
North-Holland Mathematics Studies , 115:173–178, 1985.[2] J. Balogh, B. Csaba, and W. Samotij. Local resilience of almost spanning trees in randomgraphs.
Random Structures & Algorithms , 38(1-2):121–139, 2011.[3] S. Ben-Shimon, M. Krivelevich, and B. Sudakov. Local resilience and Hamiltonicity Maker–Breaker games in random regular graphs.
Combinatorics, Probability and Computing ,20(02):173–211, 2011.[4] S. Ben-Shimon, M. Krivelevich, and B. Sudakov. On the resilience of Hamiltonicity andoptimal packing of Hamilton cycles in random graphs.
SIAM Journal on Discrete Mathe-matics , 25(3):1176–1193, 2011.[5] B. Bollob´as. The evolution of sparse graphs. In
Graph theory and combinatorics (Cambridge,1983) , pages 35–57. Academic Press, London, 1984.[6] B. Bollob´as. Random graphs. In
Modern Graph Theory , pages 215–252. Springer, 1998.[7] B. Bollob´as and A. Thomason. Random graphs of small order.
North-Holland MathematicsStudies , 118:47–97, 1985.[8] J. B¨ottcher, Y. Kohayakawa, and A. Taraz. Almost spanning subgraphs of random graphsafter adversarial edge removal.
Combinatorics, Probability and Computing , 22(5):639–683,2013.[9] J. B¨ottcher, M. Schacht, and A. Taraz. Proof of the bandwidth conjecture of Bollob´as andKoml´os.
Mathematische Annalen , 343(1):175–205, 2009.[10] R. Diestel.
Graph theory . Springer-verlag, 2000.[11] G. A. Dirac. Some theorems on abstract graphs.
Proceedings of the London MathematicalSociety , 3(1):69–81, 1952.[12] P. Erd˝os and A. R´enyi. On the existence of a factor of degree one of a connected randomgraph.
Acta Math. Acad. Sci. Hungar. , 17:359–368, 1966.2113] P. Erd˝os and A. R´enyi. On random graphs I.
Publicationes Mathematicae Debrecen , 6:290–297, 1959.[14] A. Frieze and M. Karo´nski.
Introduction to random graphs . Cambridge University Press,2015.[15] A. Frieze and M. Krivelevich. On two Hamilton cycle problems in random graphs.
IsraelJournal of Mathematics , 166(1):221–234, 2008.[16] E. N. Gilbert. Random graphs.
The Annals of Mathematical Statistics , 30(4):1141–1144,1959.[17] A. Hajnal and E. Szemer´edi. Proof of a conjecture of P. Erd˝os.
Combinatorial theory andits applications , 2:601–623, 1970.[18] H. Huang, C. Lee, and B. Sudakov. Bandwidth theorem for random graphs.
Journal ofCombinatorial Theory, Series B , 102(1):14–37, 2012.[19] S. Janson, T. (cid:32)Luczak, and A. Ruci´nski.
Random graphs , volume 45. John Wiley & Sons,2011.[20] J. Koml´os, G. N. S´ark¨ozy, and E. Szemer´edi. On the P´osa-Seymour conjecture.
Journal ofGraph Theory , 29(3):167–176, 1998.[21] J. Koml´os and E. Szemer´edi. Limit distribution for the existence of Hamiltonian cycles ina random graph.
Discrete Mathematics , 43(1):55–63, 1983.[22] M. Krivelevich. Long paths and Hamiltonicity in random graphs. arXiv preprintarXiv:1507.00205 , 2015.[23] M. Krivelevich, C. Lee, and B. Sudakov. Resilient pancyclicity of random and pseudorandomgraphs.
SIAM Journal on Discrete Mathematics , 24(1):1–16, 2010.[24] M. Krivelevich, C. Lee, and B. Sudakov. Robust Hamiltonicity of Dirac graphs.
Transac-tions of the American Mathematical Society , 366(6):3095–3130, 2014.[25] M. Krivelevich and B. Sudakov. Pseudo-random graphs.
More sets, graphs and numbers ,pages 199–262, 2006.[26] D. K¨uhn and D. Osthus. The minimum degree threshold for perfect graph packings.
Com-binatorica , 29(1):65–107, 2009.[27] C. Lee and B. Sudakov. Dirac’s theorem for random graphs.
Random Structures & Algo-rithms , 41(3):293–305, 2012.[28] T. (cid:32)Luczak. On matchings and Hamiltonian cycles in subgraphs of random graphs.
North-Holland Mathematics Studies , 144:171–185, 1987.[29] T. (cid:32)Luczak and A. Ruci´nski. Tree-matchings in graph processes.
SIAM Journal on DiscreteMathematics , 4(1):107–120, 1991.[30] R. Montgomery. Hamiltonicity in random graphs is born resilient. arXiv preprintarXiv:1710.00505 , 2017.[31] L. P´osa. Hamiltonian circuits in random graphs.
Discrete Mathematics , 14(4):359–364,1976.[32] B. Sudakov. Robustness of graph properties.
Surveys in Combinatorics 2017 , 440:372, 2017.2233] B. Sudakov and V. H. Vu. Local resilience of graphs.