Dirac s theorem for random graphs

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1 Dirac s theorem for radom graphs Choogbum Lee Bey Sudakov Abstract A classical theorem of Dirac from 1952 asserts that every graph o vertices with miimum degree at least /2 is Hamiltoia. I this paper we exted this result to radom graphs. Motivated by the study of resiliece of radom graph properties we prove that if p log /, the a.a.s. every subgraph of G, p with miimum degree at least 1/2+o1p is Hamiltoia. Our result improves o previously kow bouds, ad aswers a ope problem of Sudakov ad Vu. Both, the rage of edge probability p ad the value of the costat 1/2 are asymptotically best possible. 1 Itroductio A Hamilto cycle of a graph is a cycle which passes through every vertex of the graph exactly oce, ad a graph is Hamiltoia if it cotais a Hamilto cycle. Hamiltoicity is oe of the most cetral otios i graph theory, ad has bee itesively studied by umerous researchers. The problem of determiig Hamiltoicity of a graph is oe of the NP-complete problems that Karp listed i his semial paper [18], ad accordigly, oe caot hope for a simple classificatio of such graphs. Therefore it is importat to fid geeral sufficiet coditios for Hamiltoicity ad i the last 60 years may iterestig results were obtaied i this directio. Oe of the first results of this type is a classical theorem proved by Dirac i 1952 see, e.g., [12, Theorem ], which asserts that every graph o vertices of miimum degree at least /2 is Hamiltoia. I this paper, we study Hamiltoicity of radom graphs. The model of radom graphs we study is the biomial model G, p also kow as the Erdős-Reyi radom graph, which deotes the probability space whose poits are graphs with vertex set [] = {1,..., } where each pair of vertices forms a edge radomly ad idepedetly with probability p. We say that G, p possesses a graph property P asymptotically almost surely, or a.a.s. for brevity, if the probability that G, p possesses P teds to 1 as goes to ifiity. The earlier results o Hamiltoicity of radom graphs were proved by Pósa [25], ad Korshuov [21]. Improvig o these results, Bollobás [10], ad Komlós ad Szemerédi [20] proved that if p log +log log +ω/ for some fuctio ω that goes to ifiity together with, the G, p is a.a.s. Hamiltoia. The rage of p caot be improved, sice if p log + log log ω/, the G, p asymptotically almost surely has a vertex of degree at most oe. Departmet of Mathematics, UCLA, Los Ageles, CA, choogbum.lee@gmail.com. Research supported i part by a Samsug Scholarship. Departmet of Mathematics, UCLA, Los Ageles, CA bsudakov@math.ucla.edu. Research supported i part by NSF grat DMS , NSF CAREER award DMS ad by USA-Israeli BSF grat. 1

2 Recetly, i [27] the authors proposed to study Hamiltoicity of radom graphs i more depth by measurig how strogly the radom graphs possess this property. Let P be a mootoe icreasig graph property. Defie the local resiliece of a graph G with respect to P as the miimum umber r such that by deletig at most r edges from each vertex of G, oe ca obtai a graph ot havig P. Usig this otio, oe ca state the aforemetioed Dirac s theorem as K has local resiliece /2 with respect to Hamiltoicity. Sudakov ad Vu [27] iitiated the systematic study of resiliece of radom ad pseudoradom graphs with respect to various properties, oe of which is Hamiltoicity. I particular, they proved that if p > log 4 /, the G, p a.a.s. has local resiliece at least 1/2 + o1p with respect to Hamiltoicity. Their result ca be viewed as a geeralizatio of Dirac s Theorem, sice a complete graph is also a radom graph G, p with p = 1. This coectio is very atural ad most of the resiliece results ca be viewed as a geeralizatio of some classic graph theory result to radom ad pseudoradom graphs. For other recet results o resiliece, see [2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 19, 22, 23, 26]. Note that i the above metioed result of Sudakov ad Vu, the costat 1/2 i the resiliece boud caot be further improved. Ideed, cosider a partitio of the vertex set of a radom graph ito two parts of size /2 ad remove all the edges betwee these parts. Sice the graph is radom this removes roughly half of the edges icidet with each vertex ad makes the graph discoected. However, thigs become uclear whe oe cosiders the rage of p. Recall that Bollobás, ad Komlós ad Szemerédi s result metioed above implies that if p > C log / for some C > 1, the G, p is a.a.s. Hamiltoia. Therefore it is atural to believe, as was cojectured i [27], that G, p has local resiliece 1/2 + o1p with respect to Hamiltoicity already whe p log /. I additio to [27], several other results have bee obtaied o this problem. Frieze ad Krivelevich [13] proved that there exist costats C ad ε such that if p C log /, the G, p a.a.s. has local resiliece at least εp with respect to Hamiltoicity. This result was improved by Be-Shimo, Krivelevich, ad Sudakov [6] who showed that for all ε, there exists a costat C, such that if p C log /, the G, p has local resiliece at least 1/6 εp. I their recet paper [7], the same authors further improved this boud to 1/3 εp. Our mai theorem completely solves the resiliece problem of Sudakov ad Vu. Theorem 1.1. For every positive ε, there exists a costat C = Cε such that for p C log, a.a.s. every subgraph of G, p with miimum degree at least 1/2 + εp is Hamiltoia. As metioed above, the costat 1/2 ad the rage of edge probability p are both asymptotically best possible. Notatio. A graph G = V, E is give by a pair of its vertex set V = V G ad edge set E = EG. We use sometimes G to deote the order of the graph. For a subset X of vertices, we use ex to deote the umber of edges withi X, ad for two sets X, Y, we use ex, Y to deote the umber of edges {x, y} such that x X, y Y ote that ex, X = 2eX. We use NX to deote the collectio of vertices which are adjacet to some vertex of X. For two graphs G 1 ad G 2 over the same vertex set V, we defie their itersectio as G 1 G 2 = V, EG 1 EG 2, their uio as G 1 G 2 = V, EG 1 EG 2, ad their differece as G 1 \ G 2 = V, EG 1 \ EG 2 Whe there are several graphs uder cosideratio, to avoid ambiguity, we use subscripts such as N G X to idicate the graph that we are curretly iterested i. We also use subscripts with asymptotic otatios to idicate depedecy. For example, Ω ε will be used to idicate that the 2

3 hidde costat depeds o ε. To simplify the presetatio, we ofte omit floor ad ceilig sigs wheever these are ot crucial ad make o attempts to optimize absolute costats ivolved. We also assume that the order of all graphs teds to ifiity ad therefore is sufficietly large wheever ecessary. 2 Properties of radom graphs I this sectio we develop some properties of radom graphs. The followig cocetratio result, Cheroff s boud see, e.g., [1, Theorem A.1.12], will be used to establish these properties. Theorem 2.1. Let ε be a positive costat. If X be a biomial radom variable with parameters ad p, the P X p εp e Ωεp. Also, for λ 3p, P X p λ e Ωλ. We first state two stadard results o radom graphs, which estimates the umber of edges ad the degree of vertices. We omit their proofs which cosist of straightforward applicatios of Cheroff s iequality. Propositio 2.2. For every positive ε, there exists a costat C such that for p C log, the radom graph G = G, p a.a.s. has eg = 1+o1 2 p 2 edges, ad v V, 1 εp degv 1 + εp. Propositio 2.3. Let p log /, ad ω be a arbitrary fuctio which goes to ifiity as goes to ifiity. The i G = G, p, a.a.s. for every two subsets of vertices X ad Y, ex, Y = X Y p + o X Y p + ω. It is well-kow that radom graphs have certai expasio properties, ad that these properties are very useful i provig Hamiltoicity. Next propositio shows that the expasio property still holds eve after removig some of its edges. Similar lemmas appeared i [22, 27]. Propositio 2.4. For every positive ε, there exists a costat C such that for p C log, the radom graph G = G, p a.a.s. has the followig property. For every graph H of maximum degree at most 1 2 εp, the graph G = G H satisfies the followig: i X V, X log 1/4 p 1, N G X ε X p, ii X V, log 1/2 X ε 4, N G X ε, ad iii G is coected. Proof. Let H be a graph of maximum degree at most 1 2 εp, ad let G = G H. i To prove i, it suffices to prove that a.a.s. for all X V of size at most log 1/4 p 1, N G X 1 ε X p, 2 3

4 sice it will imply by the maximum degree coditio of H that 1 N G X N G X 2 ε p X ε 2 p X. Fix a set X V of size X log 1/4 p 1. For each v V \ X let Y v be idicator radom variable of the evet that v NX. We have P Y v = 1 = 1 1 p X = 1 + o1 X p the estimate follows from the fact X p = o1. Cosider the radom variable Y = v V \X Y v = NX \ X ad ote that E[Y ] = P Y v = 1 = X 1 + o1 X p = 1 + o1 X p. v V \X Sice the evets Y v are mutually idepedet, we ca apply the Cheroff s iequality to get P Y E[Y ] ε/3e[y ] e ΩεE[Y ]. Combie this with the estimate o E[Y ] ad we have, P Y 1 ε/2 X p e ΩεE[Y ] = e Ωε X p. for large eough. Sice Y NX ad p C log, the probability that NX < 1 ε 2 X p is e Ωε X p = C X, where C = C ε, C ca be made arbitrarily large by choosig costat C appropriately. Takig the uio boud over all choices of X, we get 1 X log 1/4 p 1 C X which establishes our claim. k=1 C k k e k C k = o1, k=1 ii We will first prove that a.a.s. for every pair of disjoit sets X, Y of sizes log 1/2 X ε 4 ad Y 1 2 3ε 4, e G X, Y 1 ε 1 X Y p > 4 2 ε X p. 1 Ideed, let X, Y be a fixed pair of disjoit sets such that log 1/2 X ε ε 4. The E[eG X, Y ] = X Y p ad by Cheroff s iequality, P e G X, Y 1 ε/4 X Y p < e Ωε X Y p e Ωεlog 1/2. ad Y Sice there are at most 2 2 possible choices of the pairs X, Y ad the probability above is 2 2, takig the uio boud will give our coclusio. Coditio o the evet that 1 holds, ad assume that there exists a set X of size log 1/2 X ε 4 which does ot have at least ε 2 eighbors i G. The there exists a set Y of size at least Y ε 2 X 1 2 3ε 4 disjoit from X such that there are o edges betwee X ad Y i G. However, this gives us a cotradictio to 1 sice 1 0 = e G X, Y e G X, Y 2 ε p X > 0. 4

5 iii Coditio o the evet that i ad ii holds, ad assume that G is ot coected. Let X be a set of vertices which iduces a coected compoet i G, ad let Y = V \ X. By part i, we kow that X log 1/4 p 1 p 2 = 1 2 log 1/4, ad the by part ii, we kow that X > 2. O the other had, sice Y must also cotai a coected compoet, the same estimate must hold for Y as well. However this caot happe sice the total umber of vertices is. Therefore, G is coected. 3 Rotatio ad extesio Our mai tool i provig Hamiltoicity is Pósa s rotatio-extesio techique see [25] ad [24, Ch. 10, Problem 20]. We start by briefly discussig this powerful tool which exploits the expasio property of a graph, i order to fid log paths ad/or cycles. Let G be a coected graph ad let P = v 0,, v l be a path o some subset of vertices of G P is ot ecessarily a subgraph of G. If {v 0, v l } is a edge of G, the we ca use it to close P ito a cycle. Sice G is coected, either the graph G P is Hamiltoia, or there exists a loger path i this graph. I the secod case, we say that we exteded the path P. Assume that we caot directly exted P as above, ad assume that G cotais a edge of the form {v 0, v i } for some i. The P = v i 1,, v 0, v i, v i+1,, v l forms aother path of legth l i G P see figure 1. We say that P is obtaied from P by a rotatio with fixed edpoit v 0, pivot poit v i, ad broke edge {v i 1, v i }. Note that after performig this rotatio, we ca ow close a cycle of legth l also usig the edge {v i 1, v l } if it exists i G P. As we perform more ad more rotatios, we will get more such cadidate edges call them closig edges. The rotatio-extesio techique is employed by repeatedly rotatig the path util oe ca fid a closig edge i the graph, thereby extedig the path. Let P be a path obtaied from P by several rouds of rotatios. A importat observatio which we later will use is that for a iterval I = v j,, v k of vertices of P 0 j < k l, if o edges of I were broke durig these rotatios, the I appears i P either exactly as it does i P, or i the reversed order. We will use rotatios ad extesio as described above to prove our mai theorem. The mai techical twist is to split the give graph ito two graphs, where the first graph will be used to perform rotatios ad the secod graph to perform extesios. Similar ideas, such as spriklig, has bee used i provig may results o Hamiltoicity of radom graphs. The oe which is closest to our implemetatio, appears i the recet paper of Be-Shimo, Krivelevich, ad Sudakov [7]. I the followig two subsectios, we prove that our radom graph ideed cotais subgraphs which ca perform these two roles of rotatio ad extesio. All the graphs that we study from ow o are defied over the same vertex set, ad we will use this fact without further metioig. v 0 v i 1 v i v l Figure 1: Rotatig a path 5

6 3.1 Rotatio Defiitio 3.1. Let δ be a positive costat. We say that a coected graph G o vertices has property REδ if the followig holds for a arbitrary path P. Either i there exists a path loger tha P i the graph G P, or ii there exists a set of vertices S P of size at least S P δ such that for every vertex v S P, there exists a set T v of size T v δ such that for every w T v, there exists a path of the same legth as P i G P which starts at v ad eds at w. Iformally, a graph has property REδ if every path is either extedable to a loger path, or ca be rotated i may differet ways. The ext lemma, which is the most crucial igrediet of our proof, asserts that we ca fid a graph with property RE ε i radom graphs eve after deletig some of its edges. Lemma 3.2. For every positive ε, there exists a costat C = Cε such that for p C log, the radom graph G = G, p a.a.s. has the followig property: for every graph H of maximum degree at most 1 2 2εp, the graph G = G H satisfies RE ε. Proof. Let C be a sufficietly large costat such that the assertios of Propositios 2.2, 2.3 a.a.s. hold, ad the assertios of Propositio 2.4 a.a.s. hold with 2ε istead of ε. Coditio o all of these evets. Let H be a subgraph of G, p which has maximum degree at most 1 2 2εp, ad let G = G H. By Propositio 2.2, we kow that G has miimum degree at least εp, ad by Propositio 2.4 iii, we kow that G is coected. We wat to show that G RE ε for all choices of H. Cosider a path P = v 0,, v l. If there exists a path loger tha P i G P, the there is othig to prove. Thus we may assume that this is ot the case. For a set Z V P, let Z + = {v i+1 v i Z} ad Z = {v i 1 v i Z}. For a vertex z, let z be the vertex i {z} ad similarly defie z +. Step 1 : Iitial rotatios. First we show that there exists a set X of liear size such that for all v X, there exists a path of legth l startig at v ad edig at v l. Such X will be costructed iteratively. I the begiig, let X 0 = {v 0 }. Now suppose that we have costructed sets X i of sizes 4 i p i up to some oegative i. If 4 i p i max{1, log 1/4 p 1 }, the either by the miimum degree of G i case, whe X i = 1 or by Propositio 2.4 i, we kow that N G X i ε X i p. We must have N G X i P as otherwise we ca fid a path loger tha P. Cosequetly, we ca rotate the edpoits X i usig the vertices i N G X i as pivot poits. If a vertex w N G X i does ot belog to ay of X j, Xj, X+ j for j < i, the both edges of the path P icidet with w were ot broke i the previous rotatios. Hece, usig w as a pivot poit, we get either w or w + as a ew edpoit see the discussio at the begiig of the sectio. Therefore, at most two such pivot poits ca give rise to a same ew edpoit, ad we obtai a set X i+1 of size at least X i+1 1 i N G X i 3 X j 2 j=0 1 1 p i ε p op i i+1 p i+1, 4 6

7 where i j=0 X j = op i+1 sice X j = p/4 j ad p C log. Repeat the argumet above util at step t we have a set of edpoits X t of size at least log 1/4 p 1, ad redefie X t by arbitrarily takig a subset of this set of size X t = max{1, log 1/4 p 1 } ote that t log logp/4 log log log for C 4. Apply the same argumet as above to X t to fid a set of { } edpoits of size at least max p 4,. Agai, if ecessary, redefie X log 1/2 t+1 to be a arbitrary subset of this set of size X t+1 = / log 1/2, ad repeat the argumet above oe more time, ow usig the secod part of Propositio 2.4 istead of the first part to get N G X t ε. I the ed, we obtai a set X t+2 of size at least X t+2 1 t+1 N G X t+1 3 X t 2 4. Step 2 : Termial rotatio. Let X = X t+2 be the set of size at least 4 that we costructed i Step 1. We will show that aother roud of rotatio gives at least ε edpoits. Let Y be the set of all edpoits that we obtai by rotatig X oe more time ote that Y ca cotai vertices from X. Partitio the path P ito k = log /log log 1/2 itervals P 1,, P k of legths as equal as possible. Every vertex w X was obtaied by t + 2 rotatios which broke t + 2 edges of P. If the iterval P i cotais oe of these edges the the path from w to v l must traverse P i exactly i the same order as i P, or i the reverse order see the discussio at the begiig of the sectio. Let ˆX i be the collectio of vertices of X which were obtaied by rotatio with some broke edges i P i. Let X i +, X i be the vertices of X such that paths from these vertices to v l traverses P i i the origial, or reverse order, respectively. We kow that X = ˆX i X i + Xi for all i. The first key observatio is that the set ˆX i is small for most idices. Let J be the collectio of idices which have ˆX i log log 1/4 X. Sice each vertex i X is obtaied by at most log log log + 2 < 2 log log log rotatios, we ca double cout the total umber of broke edges used for costructig all the poits of X to get j=1 J log log 1/4 X X 2 log log log, which implies J 2 log /log log 3/4 = ok. Our secod key observatio comes from the fact that for a vertex v j P i ad a vertex x X i +, if {x, v j+1 } is a edge of G, the v j Y similarly, for x Xi, if {x, v j 1} is a edge of G, the v j Y. Therefore, for all i, there are o edges of G betwee X i + ad P i P i + \ Y +, ad betwee X i ad P i P i \ Y. We will show that if Y < 1 2 will have to remove too may edges icidet to X from the graph G. The umber of edges icidet to X that we eed to remove is at least, e G X, V \ P + k i=1 + ε, the this caot happe because we e G Xi, P i Pi \ Y + e G X i +, P i P i + \ Y +. Sice P i Pi \ Y ad P i \ Y differs by at most oe elemet similar for P i +, the above 7

8 expressio is e G X, V \ P + k i=1 e G Xi, P i \ Y + e G X i +, P i \ Y + + O. By defiitio, P i = P /k = O log log log 1/2 ad X i = O. Thus, we ca use Propositio 2.3 to get X V \ P p + o 2 p+ k i=1 X i P i \ Y p + X + i P i \ Y + p + o 2 p log log 1/2. log Sice X = ˆX i X + i X i ad P i \ Y P i \ Y 1 also for P i \ Y +, this equals to k X V \ P p + X \ ˆX i P i \ Y p + o 2 p. i=1 As observed above, X \ ˆX i = 1 o1 X for all but ok of idices i, ad hece this expressio becomes X V \ P p + X p k i=1 P i \ Y ok P k X p + o2 p = X V \ Y p + o 2 p. O the other had, this is at most the umber of edges icidet with X i the graph H which we removed, so it must be less tha X 1 2 2εp. Sice X /4, we must have V \ Y 1 2 2ε + o1 ad therefore Y ε. Step 3 : Rotatig the other edpoit. I Steps 1 ad 2, we costructed a set S P of size S P ε such that for all v S P, there exists a path of legth l which starts at v ad eds at v l. For each of these paths, we do the same process as i Steps 1 ad 2, ow keepig v fixed ad rotatig the other edpoit v l. I this way we ca costruct the sets T v required for the property RE ε. 3.2 Extesio I the previous subsectio, we showed that radom graphs cotai subgraphs which ca be used to perform the role of rotatios. I this subsectio, we show that there exist graphs which ca perform the role of extesios. Defiitio 3.3. Let δ be a positive costat ad let G 1 be a graph o vertices with property REδ. We say that a graph G 2 complemets G 1, if for every path P, either there exists a path loger tha P i G 1 P, or there exist vertices v S P ad w T v such that {v, w} is a edge of G 1 G 2 the sets S P ad T v are defied as i Defiitio 3.1. Propositio 3.4. Let δ be a fixed positive costat. For every G 1 REδ ad G 2 complemetig G 1, the uio G 1 G 2 is Hamiltoia. 8

9 Proof. Let P be the logest path i G 1 G 2. By the defiitio of REδ, there exists a set S P such that for all v S P, there exists a set T v such that for all w T v, there exists a path of the same legth as P which starts at v ad eds at w. By the defiitio of G 2, there exists v S P ad w T v such that {v, w} is a edge of G 1 G 2. Therefore we have a cycle of legth P i G 1 G 2. Either this cycle is a Hamilto cycle or it is discoected to the rest of the graph, as otherwise it cotradicts the assumptio that P is the logest path. However, the latter caot happe sice the graph G 1 is coected by the defiitio of REδ. Thus we ca coclude that the cycle we foud is ideed a Hamilto cycle. The ext lemma is the mai lemma of this subsectio ad says that the radom graph complemets all of its subgraphs with small umber of edges. Lemma 3.5. For every fixed positive ε, there exist costats δ = δε ad C = Cε such that G = G, p a.a.s. has the followig property: for every graph H of maximum degree at most 1 2 εp, the graph G = G H complemets all subgraphs R G which satisfy RE ε ad have at most δ 2 p edges. Proof. Let G be some subgraph of G obtaied by removig at most 1 2 εp edges icidet to each vertex. The probability that the assertio of the lemma fails is P = P {R G} {some G does ot complemet R} 2 R RE 1 2 +ε, ER δ2 p R RE 1 2 +ε, ER δ2 p P some G does ot complemet R R G PR G, where the uio ad sum is take over all labeled graphs R o vertices which has property RE ε ad at most δ2 p edges. Let us first examie the term P some G does ot complemet R R G. Let R be a fixed graph with property RE ε, ad P be a fixed path o the same vertex set. The umber of such paths is at most!, sice there are choices for the legth of path P ad there are at most 1 i + 1 paths of legth i, 1 i. If i R P there is a path loger tha P, the the coditio of Defiitio 3.3 is already satisfied. Therefore we ca assume that there is o such path i R P. The, by the defiitio of property RE ε, we ca fid a set S P ad for every v S P a corespodig set T v, both of size ε, such that for every w T v, there exists a path of the same legth as P i R P which starts at v ad eds at w. If there exists a vertex v S P ad w T v such that {v, w} is a edge of R, the this edge is also i R G ad agai Defiitio 3.3 is satisfied. If there are o such edges of R, the sice R is a labeled graph, coditioed o R G, each such pair of vertices is a edge i G idepedetly with probability p. Let S P be a arbitrary subset of S P of size ε 2, ad for each v S P, defie T v to be the set T v \ S P. Sice T v ε 2, by Cheroff s iequality, for a fixed vertex v S P, the probability that i G, p this vertex has less tha 1 2 p eighbors i T v is at most e Ωεp. Sice S P is disjoit from all the sets T v, these evets are idepedet for differet vertices. Thus, usig that S P = ε 2, we ca see that the probability that all vertices v S P have less tha 1 2 p eighbors i T v is at most e Ωε2p. 9

10 Note that if some vertex v S P has at least 1 2 p eighbors i T v, the sice G was obtaied from G by removig at most 1 2 εp edges from each vertex, there must be a vertex w T v such that {v, w} is a edge i G. Therefore if some G does ot complemet the graph R, the a.a.s. there exists some path P such that all vertices v S P have less tha 1 2 p eighbors i T v. Takig the uio boud over all choice of paths P, we see that for large eough C = Cε ad p C log P some G does ot complemet R R G! e Ωε2p = e Ωε2p. Therefore i 2, the right had side ca be bouded by P e Ωε2 p R RE 1 2 +ε, ER δ2 p PR G. Also ote that for a fixed labeled graph R with k edges PR G, p = p k. Therefore, by takig the sum over all possible graphs R with at most δ 2 p edges, we ca boud the probability that the assertio of the lemma fails by P e Ωε2 p δ 2 p k=1 2 k p k e Ωε2 p δ 2 p k=1 e 2 p k. For δ 1, the summad is mootoe icreasig i the rage 1 k δ 2 p, ad thus we ca take the case k = δ 2 p for a upper boud o every term. This gives P e Ωε2 p δ 2 p eδ 1 δ 2 p = e Ω ε 2p e Oδ log1/δ2p, k which is o1 for sufficietly small δ depedig o ε. This completes the proof. 4 Proof of the mai theorem I this sectio we prove the mai theorem. I view of Lemmas 3.2 ad 3.5, we ca fid both the graphs we eed to perform rotatios ad extesios. However, we caot aively apply the two lemmas together, sice i order to have valid extesios i Lemma 3.5, we eed the rotatio graph to have at most δ 2 p edges. Thus to complete the proof, we fid a rotatio graph which has at most δ 2 p edges. Before proceedig, we state aother useful cocetratio result see, e.g., [17, Theorem 2.10]. Let A ad A be sets such that A A. Let B be a fixed size subset of A chose uiformly at radom. The the distributio of the radom variable B A is called the hypergeometric distributio. Theorem 4.1. Let ε be a fixed positive costat ad let X be a radom variable with hypergeometric distributio. The, P X E[X] εe[x] e ΩεE[X]. 10

11 Lemma 4.2. For every positive ε ad δ < 1, there exists a costat C = Cε, δ such that for p C log, the radom graph G, p a.a.s. has the followig property. For every graph H of maximum degree at most 1 2 3εp, the graph G = G, p H cotais a subgraph with at most δ 2 p edges satisfyig RE ε. Proof. Let C be a sufficietly large costat such that for p C log assertios of Propositio 2.2 ad Lemma 3.2 hold a.a.s. ad let C = C /δ. Let p = δp ad let Ĝ be the graph obtaied from G, p by takig every edge of G idepedetly with probability δ. We wat to aalyze two properties of Ĝ which together will imply our claim. Call Ĝ good if it has at most 2 p = δ 2 p edges, ad all of its subgraphs obtaied by removig at most 1 2 2εp edges icidet to each vertex satisfy RE ε. Otherwise call it bad. Note that, by defiitio, the edge distributio of Ĝ is idetical to that of G, p, ad therefore by Propositio 2.2 ad Lemma 3.2, the probability that Ĝ is good is 1 o1. Let P be the collectio of graphs G for which PĜ is good G, p = G 3 4. Sice o1 = PĜ is bad PG, p / P PĜ is bad G, p / P 1 PG, p / P, 4 we kow that PG, p / P = o1, or i other words, PG, p P = 1 o1. Thus from ow o, we coditio o the evet that G, p P. Let H be a graph over the same vertex set as G, p which has maximum degree at most 1 2 3εp. Usig the cocetratio of hypergeometric distributio ad takig uio boud over all vertices of H, we have that with probability 1 o1 the graph Ĝ H has maximum degree at most 1 2 2εp. For a arbitrary choice of H, sice Ĝ is good with probability at least 3 4, ad Ĝ H has maximum degree at most 1 2 2εp with probability 1 o1, there exists a choice of Ĝ which satisfies these two properties. For such Ĝ, by the defiitio of good, the graph Ĝ H satisfies RE ε. Moreover, Ĝ has at most δ 2 p edges ad hece so does Ĝ H. Sice Ĝ H G, p H, this proves the claim. The mai result of the paper easily follows from the facts we have established so far. Proof of Theorem 1.1. Let δ be sufficietly small ad C be sufficietly large costats such that the radom graph G = G, p with p C log a.a.s. satisfies Propositio 2.2 with ε/2 istead of ε, ad the assertios of Lemmas 3.5 ad 4.2 with ε/6 istead of ε. Coditio o these evets. By Propositio 2.2, G, p has maximum degree at most 1+ ε 2 p, ad thus every subgraph of G, p of miimum degree at least εp ca be obtaied by removig a graph H of maximum degree at most 1 2 ε 2 p. Thus it suffices to show that for every graph H o vertices with maximum degree at most 1 2 ε 2 p, the graph G, p H is Hamiltoia. Let H be a graph as above. By Lemma 4.2, there exists a subgraph of G, p H which has at most δ 2 p edges ad has property RE ε 6. By Lemma 3.5, G, p H complemets this subgraph. Therefore, by Propositio 3.4, G, p H is Hamiltoia. 5 Cocludig remarks I this paper, we proved that whe p log /, every subgraph of the radom graph G, p with miimum degree at least 1/2 + o1p is Hamiltoia. This shows that G, p has local resiliece 11

12 1/2 + o1p with respect to Hamiltoicity ad positively aswers the questio of Sudakov ad Vu. It would be very iterestig to better uderstad the resiliece of radom graphs for values of edge probability more close to log /, which is a threshold for Hamiltoicity. To formalize this questio we eed some defiitios from [7]. Let a = a 1,..., a ad b = b 1,..., b be two sequeces of umbers. We write a b if a i b i for every 1 i. Give a labeled graph G o vertices we deote its degree sequece by d G = d 1,..., d. Defiitio 5.1. Let G = [], E be a graph. Give a sequece k = k 1,..., k ad a mootoe icreasig graph property P, we say that G is k-resiliet with respect to the property P if for every subgraph H G such that d H k, we have G H P. It is a itriguig ope problem to get a good characterizatio of sequeces k such the radom graph G, p with p close to log / is k-resiliet with respect to Hamiltoicity. Some results i this directio were obtaied i [7]. Ackowledgmet. We would like to thak Michael Krivelevich for ispirig discussios ad coversatios. We would also like to thak Ala Frieze for his valuable remarks. Refereces [1] N. Alo ad J. Specer, The Probabilistic Method, 2d ed., Wiley, New York, [2] N. Alo ad B. Sudakov, Icreasig the chromatic umber of a radom graph, Joural of Combiatorics , [3] J. Balogh, B. Csaba, ad W. Samotij, Local resiliece of almost spaig trees i radom graphs, Radom Structures ad Algorithms , [4] J. Balogh, C. Lee, W. Samotij, Corrádi ad Hajal s theorem for sparse radom graphs, arxiv: v1 [math.co]. [5] I. Be-Eliezer, M. Krivelevich ad B. Sudakov, Log cycles i subgraphs of pseudoradom directed graphs, J. Graph Theory, to appear. [6] S. Be-Shimo, M. Krivelevich, ad B. Sudakov, Local resiliece ad Hamiltoicity Maker- Breaker games i radom-regular graphs, Combiatorics, Probability, ad Computig , [7] S. Be-Shimo, M. Krivelevich, ad B. Sudakov, O the resiliece of Hamiltoicity ad optimal packig of Hamilto cycles i radom graphs, SIAM J. of Discrete Math , [8] J. Böttcher, Y. Kohayakawa, ad A. Taraz, Almost spaig subgraphs of radom graphs after adversarial edge removal, arxiv: v1 [math.co]. [9] D. Colo ad T. Gowers, Combiatorial theorems relative to a radom set, arxiv: v1 [math.co]. 12

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