Hamiltonicity of graphs perturbed by a random geometric graph
aa r X i v : . [ m a t h . C O ] F e b HAMILTONICITY OF GRAPHS PERTURBED BY A RANDOM GEOMETRIC GRAPH
ALBERTO ESPUNY D´IAZA
BSTRACT . We study Hamiltonicity in graphs obtained as the union of a deterministic n -vertexgraph H with linear degrees and a d -dimensional random geometric graph G d ( n , r ) , for any d ≥
1. We obtain an asymptotically optimal bound on the minimum r for which a.a.s. H ∪ G d ( n , r ) is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in suchgraphs.
1. I
NTRODUCTION
Randomly perturbed graphs are one of the thriving areas in the study of random combin-atorial structures, and many interesting results in this field have been proved in recent years.The main general goal in this area is to study properties of graphs which are obtained as theunion of a deterministic graph H (usually with a minimum degree condition) and a randomgraph G , particularly when H does not satisfy the property of interest and G is unlikely to.Research in this direction sparked off with the work of Bohman, Frieze and Martin [4],who studied Hamiltonicity (that is, the property of containing a cycle which covers everyvertex of a graph) in H ∪ G n , p , where H is an n -vertex graph with minimum degree at least α n , for some α ∈ (
0, 1/2 ) (if α ≥ H contains aHamilton cycle), and G n , p is the binomial random graph where each edge is included inde-pendently with probability p . They showed that, whenever p ≥ C ( α ) / n , asymptotically al-most surely (a.a.s.) H ∪ G n , p is Hamiltonian, improving on the threshold for Hamiltonicity in G n , p by a logarithmic factor. Since then, Hamiltonicity has also been considered in randomlyperturbed directed graphs [4, 15], hypergraphs [12, 15, 17] and subgraphs of the hypercube [7].Many other properties have been considered as well (e.g., powers of Hamilton cycles [1, 6, 8], F -factors [3, 11], spanning trees [5, 14, 16] or general bounded degree spanning graphs [6]), andin all cases significant improvements on the probability threshold have been achieved. To thebest of our knowledge, all of these results consider (hyper/di)graphs perturbed by a binomialrandom structure, such as G n , p , or its G n , m counterpart. Only very recently, Espuny D´ıaz andGir˜ao [10] considered Hamiltonicity in graphs perturbed by a random regular graph.In this paper we consider graphs perturbed by a random geometric graph. We consider(labelled) random geometric graphs in d dimensions, which are defined in the following way.Let V : = {
1, . . . , n } , and let X , . . . , X n be n independent uniform random variables on [
0, 1 ] d .Then, consider a positive real number r and let E : = {{ i , j } : k X i − X j k ≤ r } , where k·k denotesthe Euclidean norm. The resulting graph ( V , E ) is denoted by G d ( n , r ) .Hamitonicity of random geometric graphs is fairly well understood. For dimension 2, D´ıaz,Mitsche and P´erez-Gim´enez [9] determined that the sharp threshold for Hamiltonicity is r ∗ : =( log n / ( π n )) (which means that, for all ε >
0, if r ≥ ( + ε ) r ∗ , then a.a.s. G ( n , r ) isHamiltonian, and if r ≤ ( − ε ) r ∗ , then a.a.s. G ( n , r ) is not Hamiltonian). This result waslater strengthened by Balogh, Bollob´as, Krivelevich, M ¨uller and Walters [2] and, independ-ently, M ¨uller, P´erez-Gim´enez and Wormald [18], who also extended their results to higherdimensions, proving that, for each positive integer d ≥
2, there exists a constant c d such thatthe sharp threshold for Hamiltonicity in G d ( n , r ) is ( c d log n / n ) d . (These two papers actually I NSTITUT F ¨ UR M ATHEMATIK , T
ECHNISCHE U NIVERSIT ¨ AT I LMENAU , 98684 I
LMENAU , G
ERMANY . E-mail address : [email protected] . Date : 5th February 2021.This research has been partially supported by the Carl Zeiss Foundation. prove a stronger “hitting time” result.) The same statement is true for d =
1, which is a specialcase (see [19, Section 13.1] and the references therein, together with a remark in the introduc-tion of [2]). All of these results also extend to other ℓ p norms, 1 ≤ p ≤ ∞ , the only differencebeing the value of the constant c d = c ( d , p ) . For more information about randon geometricgraphs, we recommend the works of Penrose [19, 20].Our goal here is to prove the following result. Theorem 1.
For every integer d ≥ and α ∈ (
0, 1/2 ) , there exists a constant C such that thefollowing holds. Let H be an n-vertex graph with minimum degree at least α n, and let r ≥ ( C / n ) d .Then, a.a.s. H ∪ G d ( n , r ) is Hamiltonian. This result is asymptotically best possible, up to the constant factor (which we will makeno effort to optimise). The rest of the paper is organised as follows. In section 2, we set ournotation and also introduce a probabilistic tool which will be important for our proof. We thenprove theorem 1 in section 3. Finally, in section 4 we make several observations about ourresult, as well as some extensions that follow from its proof.2. P
RELIMINARIES
Notation.
For any positive integer n , we denote [ n ] : = {
1, . . . , n } . If we write parametersin a hierarchy, we assume they are chosen from right to left. To be more precise, wheneverwe write 0 < a ≪ b ≤
1, we mean that there exists an unspecified, non-decreasing function f : R → R such that the ensuing claim holds for all 0 < b ≤ < a ≤ f ( b ) . This canbe immediately generalised to longer hierarchies, and also to hierarchies where one parametermay depend on two or more other parameters. We say that a sequence of events {E i } i ∈ N holds asymptotically almost surely (a.a.s.) if P [ E i ] → i → ∞ . In all asymptotic statements, we willignore rounding issues whenever these do not affect the arguments.Most of our graph theoretic notation is standard. Given a graph G , we use V ( G ) and E ( G ) todenote its vertex set and edge set, respectively. We always consider labelled graphs, meaningthat whenever we say that G is an n -vertex graph we may implicitly assume that V ( G ) = [ n ] .If G is a geometric graph (meaning here that each of its vertices is assigned to a position in R d for some integer d ), then V ( G ) may interchangeably be used to refer to the set of positionsto which the vertices of G are assigned, and similarly the notation v may refer to a vertex orto its position. If e = { u , v } is an edge, we usually abbreviate it as e = uv . Given any vertex v ∈ V ( G ) , we define N G ( v ) : = { u ∈ V ( G ) : uv ∈ E ( G ) } , and d G ( v ) : = | N G ( v ) | is its degree . Wedenote the minimum and maximum vertex degrees of G by δ ( G ) and ∆ ( G ) , respectively. Givena graph G and a set of vertices A ⊆ V ( G ) , we denote by G [ A ] the graph on vertex set A whoseedges are all edges of G which have both endpoints in A . A path P is a graph whose verticescan be labelled in such a way that E ( P ) = { v i v i + : i ∈ [ | V ( P ) | − ] } . If the endpoints of a path(the first and last vertices in the labelling described above) are u and v , we sometimes refer toit as a ( u , v ) -path. Given a ( u , v ) -path P and a ( v , w ) -path P ′ such that V ( P ) ∩ V ( P ′ ) = { v } ,we write PP ′ to denote the path obtained by concatenating P and P ′ (formally, this is the uniongraph of P and P ′ ). If P ′ is a single edge vw , we will write this as Pw . Multiple concatenationswill be written in the same way.2.2. Azuma’s inequality.
Let Ω be an arbitrary set (in our case, we will take Ω = [
0, 1 ] d ),and let f : Ω n → R be some function. We say that f is L -Lipschitz, for some positive L ∈ R ,if, for all x , y ∈ Ω n such that x and y are identical in all but one coordinate, we have that | f ( x ) − f ( y ) | ≤ L . The following consequence of Azuma’s inequality (see, for instance, thebook of Janson, Łuczak and Ruci ´nski [13, Corollary 2.27]) will be useful for bounding thedeviations of certain random variables. Lemma 2.
Let X , . . . , X n be independent random variables taking values in a set Ω , and let f : Ω n → R be an L-Lipschitz function. Then, for any t ≥ , the random variable X = f ( X , . . . , X n ) satisfiesthat P [ X ≥ E [ X ] + t ] ≤ e − t L n and P [ X ≤ E [ X ] − t ] ≤ e − t L n . AMILTONICITY OF GRAPHS PERTURBED BY A RANDOM GEOMETRIC GRAPH 3
3. P
ROOF OF THEOREM < n ≪ C ′ ≪ d , α ≤
1, where d ∈ N and C ′ plays the role of the constant C in the statement. Throughout, we assume that n ≥ n . Let r : = ( C ′ / n ) d , s : = ⌈ √ d / r ⌉ and C : = s d n . Partition [
0, 1 ] d into d -dimensional hypercubes of side s (intuitively, s is closeto r / ( √ d ) , only slightly smaller to guarantee that the partition above exists, and our choiceof C ′ ensures that C is sufficiently large for all the ensuing claims to hold). We refer to eachof the smaller d -dimensional hypercubes as a cell , and denote the set of all cells as C . We saythat two cells c , c ∈ C are friends if there exists a third cell whose boundary intersects thoseof c and c . It follows that each cell is friends with at most 5 d − S ⊆ [
0, 1 ] d , we say that a cell is sparse in S if it contains at most 2 · d points, and we callit dense in S otherwise.Consider a labelling of the vertices of V ( H ) as v , . . . , v n , and let X , . . . , X n be independentuniform random variables on [
0, 1 ] d . Consider the random geometric graph G = G d ( n , r ) onthe vertex set of H obtained when assigning position X i to v i . Note that, by the definition of s ,if some v ∈ V ( H ) lies in a cell c ∈ C , then it is joined by an edge of G to all other verticesin c , as well as to all vertices in cells which are friends of c . For any vertex v ∈ V ( H ) , wesay that a cell is v-sparse (in V ( H ) with respect to H ) if it contains at most one neighbour of v in H . Otherwise, we say that it is v-dense . Similarly, given any pair of (not necessarily distinct)vertices u , v ∈ V ( H ) , we say that a cell is { u , v } -sparse (in V ( H ) with respect to H ) if it does notcontain two distinct vertices x , y ∈ V ( H ) \ { u , v } such that ux , vy ∈ E ( H ) ; if it does containsuch a pair, we say that it is { u , v } -dense . Observe that a cell is { v , v } -sparse if and only if it is v -sparse. Claim 1.
The following properties hold a.a.s.: ( i ) The number of cells which are sparse in V ( H ) is at most e − C /2 n. ( ii ) For each pair of (not necessarily distinct) vertices u , v ∈ V ( H ) , the number of { u , v } -sparsecells is at most e − α C /4 n.Proof. ( i ) Let f : ([
0, 1 ] d ) n → Z ≥ be a function which, given a set S of points x , . . . , x n ∈ [
0, 1 ] d ,returns the number of cells which are sparse in S . Note that f is a 1-Lipschitz function.For each v ∈ V ( H ) and each cell c , we have that P [ v ∈ c ] = s d . Thus, since the variables X i are independent, for a fixed cell c we have that P [ c is sparse ] = · d ∑ i = (cid:18) ni (cid:19) s di ( − s d ) n − i ≤ C · d e − C .Let Y : = f ( X , . . . , X n ) be the number of sparse cells, so E [ Y ] ≤ C · d − e − C n . Since f is1-Lipschitz, it follows by lemma 2 that P [ Y ≥ e − C /2 n ] ≤ P [ Y ≥ C · d − e − C n ] ≤ e − Θ ( n ) . ( ii ) We proceed in a similar way. For each positive integer m , let g m : ([
0, 1 ] d ) m → Z ≥ be afunction which, given a set S of m points x , . . . , x m ∈ [
0, 1 ] d , returns the number of cells whichcontain at most one point. Clearly, g m is 1-Lipschitz for every m ∈ N . Given any set V ⊆ V ( H ) ,we say that a cell is V-sparse if it contains at most one of the vertices of V .Fix two (not necessarily distinct) vertices u , v ∈ V ( H ) , and let Z be the number of { u , v } -sparse cells. Let N : = N H ( u ) ∩ N H ( v ) . We split the analysis into two cases.Assume first that ℓ : = | N | ≥ α n /2. Let i , . . . , i ℓ be the indices of the vertices of H which liein N . It follows that, for a fixed cell c , P [ c is N -sparse ] = ( − s d ) ℓ + ℓ s d ( − s d ) ℓ − ≤ Ce − α C /4 .Let Z ′ : = g ℓ ( X i , . . . , X i ℓ ) , so E [ Z ′ ] ≤ e − α C /4 n . Note that every { u , v } -sparse cell is N -sparse,so Z ≤ Z ′ . Since g ℓ is 1-Lipschitz, by lemma 2 we conclude that P [ Z ≥ e − α C /2 n ] ≤ e − Θ ( n ) . A. ESPUNY D´IAZ
Assume now that ℓ < α n /2 and let S : = N H ( u ) \ ( N H ( v ) ∪ { v } ) and T : = N H ( v ) \ ( N H ( u ) ∪{ u } ) . Note that | S | , | T | ≥ α n /2. By following the same argument as above, we have that,with probability at least 1 − e − Θ ( n ) , the number of S -sparse cells is at most 2 e − α C /4 n . The sameholds for the number of T -sparse cells. Note that every { u , v } -sparse cell must be S -sparse or T -sparse. We thus conclude that P [ Z ≥ e − α C /4 n ] ≤ e − Θ ( n ) .Finally, the statement holds by a union bound over all pairs of vertices { u , v } ⊆ V ( H ) . ◭ Condition on the event that G satisfies the properties of the statement of claim 1, whichholds a.a.s. Let C s be the set of cells which are sparse in V ( H ) , and let C d : = C \ C s . We definean auxiliary graph Γ with vertex set C d where two cells are joined by an edge whenever theyare friends. In particular, ∆ ( Γ ) ≤ d − Claim 2.
The number of connected components of Γ is at most e − C /3 n.Proof. By claim 1 ( i ) , there are at most e − C /2 n sparse cells. Assume that Γ has at least twocomponents. Given an arbitrary component of Γ with fewer than s − d /2 cells, the numberof sparse cells which are friends with some cell of this component is at least 3 d −
1. On theother hand, trivially no sparse cell can be friends with more than 5 d − Γ . A double counting argument then guarantees that the number ofcomponents is at most 2 d e − C /2 n < e − C /3 n . ◭ We are now going to construct a Hamilton cycle in H ∪ G . Roughly speaking, for eachconnected component of Γ , we will find a cycle in G spanning all vertices which lie in the cellsof this component. Then, by using some edges of H , we will incorporate all leftover verticesinto said cycles and combine the cycles into a single spanning cycle.We begin by setting up some notation. Let C ⊆ C s be the set of all cells which contain novertices of V ( H ) , and let C ⊆ C s be the set of all cells which contain exactly one vertex of V ( H ) . Let C : = C s \ ( C ∪ C ) , and C ∗ : = C ∪ C . Given any cell c ∈ C d , let Γ ( c ) be theconnected component of Γ which contains c . Let F : = C s and F ∗ : = { c ∈ C d : | V ( Γ ( c )) | = } .By claim 1 ( i ) and claim 2 we have that |F ∪ F ∗ | ≤ e − C /3 n . Both F and F ∗ constitute sets of“forbidden” cells which we will avoid when connecting vertices from different cells via edgesof H . These sets will be updated as we choose edges of H to construct the Hamilton cycle.Indeed, as we choose edges of H , each cell which contains a vertex incident to any of theseedges will be added to F . These edges of H are chosen “from a cell”, and F ∗ will contain alldense cells from which we must choose edges in a “correct” way (namely, ensuring that ourprocess works), and thus will be avoided when choosing edges of H from other cells. We claimthat, for the rest of the proof, we will always have |F ∪ F ∗ | ≤ e − C /6 n , (3.1)and assume so throughout. This bound will follow from the fact that we update F and F ∗ atmost 2 e − C /3 n times, and each time the size of their union will increase by at most 3.We first define some absorbing paths which will be used to incorporate all vertices in sparsecells into a Hamilton cycle. We define these iteratively in |C ∗ | steps. We proceed as follows.1. For each c ∈ C , let v c be the vertex contained in c , and choose an arbitrary v c -dense cell c ′ ( c ) ∈ C \ ( F ∪ F ∗ ) ; note that such a cell exists by claim 1 ( ii ) and (3.1). Choose twodistinct vertices x c , y c ∈ N H ( v c ) ∩ c ′ ( c ) . Note that e c : = x c y c ∈ E ( G ) , and define P c : = x c v c y c . Then, add c ′ ( c ) to F . Moreover, if | V ( Γ ( c ′ ( c ))) \ F | =
1, add this remaining cellto F ∗ .2. For each c ∈ C , let u c and v c be two arbitrary distinct vertices in c and choose anarbitrary { u c , v c } -dense cell c ′ ( c ) ∈ C \ ( F ∪ F ∗ ) , which again exists by claim 1 ( ii ) and (3.1). Let x c , y c ∈ ( N H ( u c ) ∪ N H ( v c )) ∩ c ′ ( c ) be two distinct vertices such that u c x c , v c y c ∈ E ( H ) , and let P ′ c be any ( u c , v c ) -path which contains all vertices in c (recallthat such a path exists because G [ V ( H ) ∩ c ] is a complete graph). Let e c : = x c y c ∈ E ( G ) AMILTONICITY OF GRAPHS PERTURBED BY A RANDOM GEOMETRIC GRAPH 5 and P c : = x c P ′ c y c . Then, add c ′ ( c ) to F . Moreover, if | V ( Γ ( c ′ ( c ))) \ F | =
1, add thisremaining cell to F ∗ .Once this process is finished, let F s : = { x c , y c : c ∈ C ∗ } and E s : = { e c : c ∈ C ∗ } .Consider an auxiliary graph Γ ′ : = Γ . We are next going to modify this graph Γ ′ into a con-nected graph. We will update Γ ′ in t − t ≤ e − C /3 n is the number of componentsof Γ . In each of these steps, we will add exactly one edge to Γ ′ , connecting two of its compon-ents. This auxiliary edge will correspond to a way in which we will later connect the cycleswhich we will construct in each component; we build the structure necessary for this at thesame time as we update Γ ′ . Our definitions of F and F ∗ are crucial in guaranteeing that theupcoming process can be carried out. In particular, F and F ∗ will always be disjoint, none ofthe components of (the current form of) Γ ′ will be contained in F , and F ∗ will always containat most one cell of each component of Γ ′ . Given any cell c ∈ C d , let Γ ′ ( c ) denote the connectedcomponent of Γ ′ which contains c . Initialise a set of vertices F d and two sets of edges E d and E ∗ d as empty sets. We proceed as follows.3. For each i ∈ [ t − ] , choose a smallest component γ of Γ ′ , and choose an arbitrary cell c ∈ V ( γ ) \ F (which exists since V ( γ ) * F ). We proceed similarly as in step 2. Let u c and v c be two arbitrary distinct vertices in c . Choose an arbitrary { u c , v c } -densecell c ′ ( c ) ∈ C \ ( F ∪ F ∗ ∪ V ( γ )) ; its existence follows from claim 1 ( ii ) , (3.1) and thefact that γ is a smallest component of Γ ′ . Add the edge { c , c ′ ( c ) } to Γ ′ . Let x c , y c ∈ ( N H ( u c ) ∪ N H ( v c )) ∩ c ′ ( c ) be two distinct vertices such that u c x c , v c y c ∈ E ( H ) . Add u c , v c , x c and y c to F d , add u c x c and v c y c to E ∗ d , and add u c v c and x c y c to E d . Then, add c and c ′ ( c ) to F (if c ∈ F ∗ , remove it from this set, so that F and F ∗ remain disjoint).Finally, if | V ( Γ ′ ( c )) \ F | =
1, add this remaining cell to F ∗ .Each iteration of step 3 reduces the number of components by one, so it follows that, after weperform all iterations, Γ ′ is connected.Let F : = F s ∪ F d and E : = E s ∪ E d . Note that, by construction, for each c ∈ C we have that | F ∩ c | ≤ | E ( G [ V ( H ) ∩ c ]) ∩ E | ≤
1. We are now ready to construct the Hamilton cycle.The main step for this is to construct a cycle in each component γ of Γ . We make sure thatthese cycles contain all edges of E spanned by the vertices in the cells of the correspondingcomponent. For each component γ of Γ , we proceed as follows.4. Let T be a spanning tree of γ . In particular, ∆ ( T ) < d . Consider an arbitrary traversalof T which, starting at a given cell, goes through every edge of T twice and ends in thestarting cell (this can be given, e.g., by a DFS on T taking any cell c as a root). Thistraversal takes m : = ( | V ( γ ) | − ) steps, each step corresponding to an edge of T . Weuse this traversal to construct a cycle C ( γ ) as follows.Assume the traversal starts in a given cell c . Choose a vertex x ∈ ( V ( H ) ∩ c ) \ F and let P : = x ; this will be the beginning of a path which we will grow into C ( γ ) .For notational purposes, set V ( P − ) : = ∅ . For each i ∈ [ m ] we define a path P i asfollows. Let c be the current cell, and let x i − ∈ ( V ( H ) ∩ c ) \ F be the last vertex of P i − . Let c ′ be the next cell of the traversal. Because c and c ′ are friends, every vertexin c ′ is joined to every vertex in c by an edge of G . Choose an arbitrary vertex x i ∈ ( V ( H ) ∩ c ′ ) \ ( F ∪ V ( P i − )) . If this is the last time that c is visited in the traversal of T ,let P ′ i be any path with vertex set ( V ( H ) ∩ c ) \ V ( P i − ) having x i − as an endpoint andsuch that, if the vertices in c span some edge e ∈ E , then e ∈ E ( P ′ i ) , and let P i : = P i − P ′ i x i ;otherwise, simply let P i : = P i − x i . To complete the cycle, let P ′ be an ( x , x m ) -pathwhich contains all vertices of ( V ( H ) ∩ c ) \ V ( P m ) and such that, if the vertices in c span some edge e ∈ E , then e ∈ E ( P ′ ) . We then set C ( γ ) : = P m ∪ P ′ . Observe thatevery cell contains at least 2 · d vertices and is visited at most 5 d times throughout thetraversal; this, together with the fact that no cell spans more than one of the edges of E ,guarantees that the choices of vertices described throughout the process can always becarried out. A. ESPUNY D´IAZ
Let C denote the graph which is the union of all the cycles constructed in step 4. In particular, E ⊆ E ( C ) . We can now combine the cycles into a single cycle spanning all vertices in cells of C d by letting C ′ : = ( C \ E d ) ∪ E ∗ d . In order to complete the proof, for each c ∈ C ∗ , replace theedge e c ∈ E ( C ′ ) by P c . (cid:3) Remark.
By using results from percolation theory, for d ≥ one can show that a.a.s. the graph Γ actually contains a “giant” component which contains, say, more than of the cells. This fact canbe used instead of claim 2 to streamline our proof, simplifying some of its technical details, such as theneed for the set F ∗ . Indeed, step 3 can be avoided entirely: for every cell c which does not lie in thisgiant component we may find a suitable { u , v } -dense cell which does, for some u , v ∈ V ( H ) ∩ c, as insteps 1 and 2. Step 4 can then be applied only to this giant component. This approach, however, doesnot work when d = , as Γ will a.a.s. not contain any component of linear size.
4. F
INAL REMARKS
Theorem 1 is asymptotically best possible in the sense that, for each α ∈ (
0, 1/2 ) , if we let r = ( c / n ) d for a sufficiently small constant c , then there exist graphs H with minimum degree α n such that H ∪ G d ( n , r ) is a.a.s. not Hamiltonian. Indeed, let H be a complete unbalancedbipartite graph with parts A and B of sizes α n and ( − α ) n , respectively. Clearly, if G d ( n , r )[ B ] contains at least n /2 isolated vertices, then H ∪ G d ( n , r ) cannot contain a Hamilton cycle. Ourclaim thus follows immediately from the following lemma. Lemma 3.
For every integer d ≥ and ε ∈ (
0, 1 ) , there exists a constant c > such that a.a.s.G d ( n , ( c / n ) d ) contains at least ( − ε ) n isolated vertices.Proof. Let 0 < c ≪ d , ε . Let f : ([
0, 1 ] d ) n → Z be a function that, given a set of n points x , . . . , x n ∈ [
0, 1 ] d , returns the number of isolated vertices in the geometric graph constructedon these points with radius ( c / n ) d . One can easily check that f is an L -Lipschitz function,for some L = L ( d ) .Fix an arbitrary i ∈ [ n ] . We are first going to compute the probability that i is isolated in G = G d ( n , ( c / n ) d ) . This is the probability that no X j with j = i is assigned to a positionwithin distance r of X i , that is, P [ i is isolated ] ≥ ( − V d c / n ) n − ≥ e − cV d ≥ − cV d ,where V d is the volume of the ball of radius 1 in d dimensions.Let X : = f ( X , . . . , X n ) denote the number of isolated vertices in G , so E [ X ] ≥ ( − cV d ) n .Since f is an L -Lipschitz function, by lemma 2 we conclude that P [ X ≤ ( − ε ) n ] ≤ P [ X ≤ ( − cV d ) n ] ≤ e − Θ ( n ) . (cid:3) We also want to remark two features of our proof of theorem 1. First, the proof is construct-ive, meaning that it provides an algorithm to find Hamilton cycles in H ∪ G d ( n , r ) . In particular,if the properties of claim 1 hold (which occurs a.a.s.), it provides a deterministic algorithm thatoutputs a Hamilton cycle in H ∪ G d ( n , r ) . Furthermore, observe that, throughout the proof, weactually do not need claim 1 ( ii ) to hold for every pair of vertices, but only for those that wepick throughout the process, which are only linearly many. This means that the properties ofclaim 1 can be checked in O ( n ) time, which is linear in the size of H ∪ G d ( n , r ) . Then, theconstruction of the Hamilton cycle also takes O ( n ) time. This follows directly from the proof,and can be checked by retracing the steps.Second, all throughout the paper we have considered the ℓ Euclidean norm for simplicity.Our proof generalises directly to all ℓ p norms, 1 ≤ p ≤ ∞ , by adjusting some of the constants.As a final remark, we note that, under the same conditions as in the statement of theorem 1,we can actually show that H ∪ G d ( n , r ) is pancyclic. Indeed, our proof can easily be modifiedfor this. From the Hamilton cycle that we construct, given that it contains many subpathswhose vertices actually from cliques in G d ( n , r ) , one can iteratively reduce the number of ver-tices in the cycle. This can be balanced with also removing some of the paths which correspond AMILTONICITY OF GRAPHS PERTURBED BY A RANDOM GEOMETRIC GRAPH 7 to sparse cells, as well as leaves of the auxiliary graph Γ ′ , to prove that cycles of all lengths canbe constructed. A CKNOWLEDGEMENT
I would like to thank Xavier P´erez-Gim´enez for some very helpful discussions about thetopic of this paper. R
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