An explicit construction of graphs of bounded degree that are far from being Hamiltonian
OOn graphs of bounded degree that are far frombeing Hamiltonian
Isolde Adler
School of Computing, University of Leeds, [email protected]
Noleen Köhler
School of Computing, University of Leeds, [email protected]
Abstract
Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an impressive amountof research has been dedicated to identifying classes of graphs that allow Hamiltonian cycles, andto related questions. The corresponding decision problem, that asks whether a given graph isHamiltonian (i. e. admits a Hamiltonian cycle), is one of Karp’s famous NP-complete problems. Itremains NP-complete on planar cubic graphs.In this paper we study graphs of bounded degree that are far from being Hamiltonian, where agraph G on n vertices is far from being Hamiltonian, if modifying a constant fraction of n edges isnecessary to make G Hamiltonian. We exhibit classes of graphs of bounded degree that are locallyHamiltonian , i.e. every subgraph induced by the neighbourhood of a small vertex set appears insome Hamiltonian graph, but that are far from being Hamiltonian.We then use these classes to obtain a lower bound in property testing. We show that in thebounded-degree graph model, Hamiltonicity is not testable with one-sided error probability andquery complexity o ( n ). This contrasts the known fact that on planar (or minor-free) graph classes,Hamiltonicity is testable with constant query complexity in the bounded-degree graph model withtwo-sided error. Our proof is an intricate construction that shows how to turn a d -regular graphinto a graph that is far from being Hamiltonian, and we use d -regular expander graphs to maintainlocal Hamiltonicity. Mathematics of computing → Paths and connectivity problems;Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords and phrases
Hamiltonian cycle, property testing, bounded-degree graphs, bounded-degreemodel, lower bound A Hamiltonian cycle in a graph G is a cycle that visits every vertex of G exactly once.A graph G is Hamiltonian , if G contains a Hamiltonian cycle. Research on Hamiltoniangraphs has a long and rich history, see e. g. [9]. Dirac’s early Theorem [4] gave sufficientconditions for Hamiltonicity, and subsequently, many further classes of Hamiltonian graphswere identified. Recently, Robinson and Wormald showed that for d ≥
3, almost all d -regulargraphs are Hamiltonian [19].Hamiltonian graphs play an important role in routing, including network design [20, 17],circuit design [21], and computer graphics [24], as well as in scheduling via tight links tothe Travelling Salesperson Problem. Deciding whether a given graph is Hamiltonian isNP-complete [12], even on cubic planar graphs [6].In this paper we study graphs of bounded degree that are far from being Hamiltonian,where intuitively, a graph G is far from being Hamiltonian if many edge modifications(insertions or deletions) are necessary to make G Hamiltonian (note that deletions may help,because of the degree bound).
On graphs of bounded degree that are far from being Hamiltonian
Motivation.
The wider motivation for our study stems from the well-known tightconnection between structural properties of graphs and their algorithmic properties, whichhas been used successfully for designing efficient algorithms for numerous problems, all theway to reaching the boundaries of efficient solvability. Hence for many important graphproperties (where by property we simply mean an isomorphism closed graph class), thestructure of graphs having the property is studied in great detail.Looking in a different direction, we are interested in the structure of graphs that are far from having a given property, where the distance is measured by the proportion of edgemodifications that are necessary to transform the graph into one that has the property.Knowing the structure of instances that are far away may help designing new, faster algorithms.This may allow to quickly distinguish between inputs that have the property, and inputsthat are far from having the property, with high probability correctly. This is the approachtaken in Property Testing. Such a distinction would allow a quick and rough first analysis ofgraphs, before using more resource intensive exact algorithms. On the other hand, a complexstructure of instances that are far away may also prohibit such an efficient distinction, andhence yield new lower bounds.We now give more details. For a given (cid:15) in the real interval [0 , G of maximum degree d with n vertices is (cid:15) -close to being Hamiltonian, if at most (cid:15)dn edgemodifications (insertions or deletions) are needed to make G Hamiltonian, and G is (cid:15) -farfrom being Hamiltonian otherwise. Note that dn is an upper bound on the total number ofedges in an n -vertex graph of degree at most d .It is easy to find graphs that are far from being Hamiltonian. For example, let G be acaterpillar graph on n = 2 k vertices as shown in Figure 1 for k = 10 (i. e. G is a path of length k where every vertex has a pendant edge). With a degree bound of at most 3, G is 1 / n/ G H consisting of k C ’s) arranged in a cycle asshown in Figure 1 for k = 9. Assume k >
1. The graph H has n = 4 k vertices and, with adegree bound of 3, H is 1 / H aHamiltonian cycle would have to traverse all four edges of every C . To avoid this we haveto increase the degree of at least one of the degree 2 vertices for every C and hence we haveto add at least n/ H Hamiltonian.In both examples it is possible to see locally, in the neighbourhood of a constant number ofvertices, that the graphs are not Hamiltonian. We ask whether there exist graphs that locallylook as if they might be Hamiltonian, but globally they are far from being Hamiltonian, andwe give a positive answer to this. More precisely, we show the following (cf. Theorem 14).
There is a d ∈ N and there are constants δ := δ ( d ) , (cid:15) := (cid:15) ( d ) ∈ (0 , and a sequence of d -bounded degree graphs ( G N ) N ∈ N of increasing order, such that G N is δ -locally Hamiltonianand (cid:15) -far from being Hamiltonian for every N ∈ N .Moreover, we give an explicit construction of the graphs G N . This has implications inproperty testing, which we explain in the next paragraph.A similar approach was taken in [3], for 3-colourability. While their main hardnessresult (in property testing) comes from a reduction from the constraint satisfaction problem(CSP), they implicitly obtain graphs which are far from being 3-colourable but locally look3-colourable, through their reduction from CSP. An explicit construction of a CSP, which isfar from being satisfiable but every sublinear subset of constraints is satisfiable, is given. Property Testing.
Property testing on graphs is a framework for studying sampling-based algorithms that solve a relaxation of classical decision problems. Given a graph G . Adler and N. Köhler 3 (a) Caterpillar (b) C ’s arranged in a cycle. Figure 1
Example graphs which are far from being Hamiltonian but are not locally Hamiltonian. and a property P (e. g. triangle-freeness), the goal of a property testing algorithm, called a property tester , is to distinguish if a graph satisfies P or is far from satisfying P , where thedefinition of far depends on the model. Property testing of dense graph is well understoodthrough its tight links with Szemeredi’s regularity Lemma [1]. In [8], Goldreich and Ronintroduced property testing on bounded-degree graphs, and since then much attention hasbeen paid to property testing in sparse graphs. Nevertheless, our understanding of testabilityof properties in such graphs is still limited. In the bounded-degree graph model [8], the testerhas oracle access to the input graph G with maximum degree d , where d is constant, andis allowed to perform neighbour queries to the oracle. That is, for any specified vertex v and index i ≤ d , the oracle returns the i -th neighbour of v if it exists or a special symbol ⊥ otherwise in constant time. A graph G with n vertices is called ε -far from satisfying aproperty P , if one needs to modify more than εdn edges to make it satisfy P . The goalnow becomes to distinguish, with probability at least 2 /
3, if G satisfies a property P oris ε -far from satisfying P , for any specified proximity parameter ε ∈ (0 , P is testable with query complexity q ( n ) in the bounded-degree model, if for every ε ∈ (0 , ε -tester ), that makes this distinction while using at most q ( n )oracle queries, where n is the size of the input graph. Property P is testable with one-sidederror , if instances in P are always correctly identified. If q is independent of n , we have constant query complexity. Here the constant can depend on ε and d . So far, it is known thatsome properties are constant-query testable, including subgraph-freeness, k -edge connectivity,cycle-freeness, being Eulerian, degree-regularity [8], minor-freeness [2, 10, 14], hyperfiniteproperties [16], k -vertex connectivity [23, 5], and subdivision-freeness [13].While the question of a full characterisation of the testable properties in the boundeddegree model is still wide open, Ito et al. [11] gave characterisations of one-sided error constant-query testable monotone graph properties, and one-sided error testable hereditary graphproperties in the bounded-degree (directed and undirected) graph model. The characterisationis based on the presence of many forbidden configurations – subgraphs in the case of monotoneproperties and induced subgraphs in the case of hereditary properties.We obtain the following lower bound for testability of Hamiltonicity (cf. Theorem 16). Hamiltonicity is not testable with one-sided error and query complexity o ( n ) in thebounded-degree model. Note that Hamiltonicity is a property that is neither monotone nor hereditary, and ourresults advance our understanding of testability of such properties. Moreover, often lowerbounds for testability are proved by randomised constructions [8, 3, 22]. We believe thatour deterministic construction enhances the understanding of the structural complexity ofinstances that are far from being Hamiltonian, which complements the rich literature on the
On graphs of bounded degree that are far from being Hamiltonian structure of graphs that are Hamiltonian.We submitted this paper for publication on the 22nd of July 2020. There, we conjecturedthat our lower bound can be strengthened to non-testability of Hamiltonicity with querycomplexity o ( n ) for two-sided error testers. Independently and simultaneously, Goldreichproved this in [7]. Gordreich’s proof uses local hardness reductions, as opposed to an explicitconstruction. We only became aware of this after our submission. Structure of the paper.
We begin with the preliminaries in Section 2. In Section 3 we introduce local
Hamiltonicity,discuss distance to Hamiltonicity, and we provide our construction. The construction takes a d -regular graph and turns it into a graph of degree at most d + 3 with additional properties.In Section 4 we prove that there is a small (cid:15) such that any family of graphs obtained viathe construction is (cid:15) -far from being Hamiltonian. Section 5 then shows that if we start ourconstruction with d -regular expander graphs, we obtain a family that is locally Hamiltonian. Let N denote the set of natural numbers including 0. We denote N ≥ n := { m ∈ N | m ≥ n } and [ n ] := { , . . . , n } for any n ∈ N .This paper concerns simple undirected graphs, however, we will use directed graphs inour construction. Unless otherwise specified graphs are undirected.An undirected graph G is a tuple ( V ( G ) , E ( G )) where V ( G ) is a finite set of vertices and E ( G ) ⊆ { e ⊆ V ( G ) | | e | = 2 } is the set of edges. A directed graph G is a tuple ( V ( G ) , E ( G ))where V ( G ) is a finite set of vertices and E ( G ) ⊆ V × V is the set of edges. For a directedgraph G and a vertex v ∈ V ( G ) we denote the set of all incoming edges of v by E − G ( v ) andthe set of all outgoing edges of v by E + G ( v ). The order of a graph G is the size of V ( G ).An isomorphism from a graph G to a graph H is a bijective map f : V ( G ) → V ( H )which preserves the edge relation, i. e., { v, w } ∈ E ( G ) iff { f ( v ) , f ( w ) } ∈ E ( H ). Equivalentlyan isomorphism from a directed graph G to a directed graph H is a bijective map f : V ( G ) → V ( H ) such that ( v, w ) ∈ E ( G ) iff ( f ( v ) , f ( w )) ∈ E ( H ). Two graphs G, H arecalled isomorphic , denoted by G ∼ = H , if there is an isomorphism between them. A graph H is a subgraph of a graph G if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ). For any graph G and S ⊆ V ( G ) we let ( S, {{ v, w } ∈ E ( G ) | v, w ∈ S } ) be the subgraph of G induced by S . Wecall a subgraph H of G an induced subgraph of G if H is the subgraph of G induced by someset S ⊆ V ( G ). For a graph G and vertices v, w ∈ V ( G ) we say that v is a neighbour of w orthat v is adjacent to w if { v, w } ∈ E ( G ). For S ⊆ V ( G ) we define the neighbourhood of S in G , denoted N G ( S ) to be the set of vertices S ∪ { v ∈ V ( G ) | v is a neighbour of some w ∈ S } .This notion of neighbourhood is often referred to as the closed neighbourhood .For a graph G (directed or undirected) the degree of a vertex v ∈ V ( G ), denoted deg G ( v ),is the number of edges that contain vertex v . The degree of a graph G , denoted deg( G ), isthe maximum degree over all vertices. A graph is called d -regular if every vertex v ∈ V ( G )has degree deg( G ), where d ∈ N . A graph has bounded degree d if deg( G ) ≤ d , where d ∈ N .For d ∈ N we denote the class of all bounded degree d graphs by C d .A path in a graph G (directed or undirected) is a sequence ( p , . . . , p ‘ ) of vertices of G such that { p i , p i +1 } ∈ E ( G )/( p i , p i +1 ) ∈ E ( G ) for i ∈ [ ‘ − simple path in G is a pathin which no vertex appears twice. A cycle is a path C = ( c , . . . , c ‘ ) such that c = c ‘ and( c , . . . , c ‘ ) is a simple path. A Hamiltonian cycle is a cycle which contains every vertex of G . We call G Hamiltonian if G contains a Hamiltonian cycle. A path P = ( p , . . . , p ‘ ) is a . Adler and N. Köhler 5 subpath of a cycle C = ( c , . . . , c ‘ ) if there is an index i ∈ [ ‘ ] such that either p j = p ( i + j mod ‘ ) for every j ∈ [ ‘ ] or p j = p ( i + ‘ − j mod ‘ ) for every j ∈ [ ‘ ]. Note that this means that subpathsappear either in the path or in the reversed path. We choose this definition of subpath forconvenient notation below.For a graph G we define the expansion ratio to be h ( G ) := min { S ⊂ V ( G ) || S |≤| V ( G ) | / } |{ e ∈ E ( G ) | | e ∩ S | = 1 }|| S | . For d ∈ N and any constant (cid:15) > G m ) m ∈ N of d -regular graphs ofincreasing number of vertices a family of (cid:15) -expanders , if h ( G m ) ≥ (cid:15) for all m ∈ N . In this section we introduce the central concepts in this paper and explain our construction.The proofs of the central properties of the construction are given in the next sections. (cid:73)
Definition 1 ( (cid:15) -farness from being Hamiltonian) . Let d ∈ N and (cid:15) ∈ [0 , . A graph G ∈ C d is (cid:15) -far from being Hamiltonian if for every set E ⊆ { e ⊆ V ( G ) | | e | = 2 } of size less or equalthen (cid:15)d · | V ( G ) | the graph ( V ( G ) , E ( G ) E ) is not Hamiltonian. (cid:73) Definition 2 (Locally Hamiltonian) . Let C be a class of graphs and let δ ∈ (0 , . Agraph G ∈ C is called δ -locally Hamiltonian on C if for every set S ⊆ V ( G ) of at most δ · | V ( G ) | vertices there is a Hamiltonian graph H := H S ∈ C with | V ( H ) | = | V ( G ) | , a subset T := T S ⊆ V ( H ) and an isomorphism from G [ N G ( S )] to H [ N H ( T )] which maps S onto T . Note that by relaxing | V ( G ) | = | V ( H ) | to | V ( H ) | > δ | V ( G ) | we get an equivalent definition. (cid:73) Remark.
Let C be a graph class. Every Hamiltonian graph in C is δ -locally Hamiltonianfor every δ ∈ (0 , G ∈ C is 1-locally Hamiltonian iff G is Hamiltonian.Let d ≥
2. A graph on n vertices is 1 /n -locally Hamiltonian on C d iff the minimum degreeof G is greater than 1.The next lemma states that if G ∈ C d has many subsets of vertices whose neighbourhoodswitness non-Hamiltonicity, then it is far from being Hamiltonian. Here we say that theneighbourhood of S ⊆ V ( G ) witnesses non-Hamiltonicity if for every Hamiltonian H ∈ C d and every T ⊆ V ( H ) there is no isomorphism from G [ N G ( S )] to H [ N H ( T )] that maps S onto T . (cid:73) Lemma 3.
Let d ∈ N ≥ and G ∈ C d . For all (cid:15) < / d , there is a number λ = λ ( (cid:15), d ) ∈ (0 , such that if there are λn subsets of V ( G ) whose (closed) neighbourhoods are pairwise disjointand each witnesses non-Hamiltonicity, then G is (cid:15) -far from being Hamiltonian. Proof.
Let λ ( (cid:15), d ) := 2 d(cid:15) . First note that if a set S ⊆ V ( G ) witnesses non-Hamiltonicity thenevery set E ⊆ { e ⊆ V ( G ) | | e | = 2 } , for which ( V ( G ) , E ( G ) E ) is Hamiltonian, must contain e such that S ∩ e = ∅ . Since the λn neighbourhoods of sets witnessing non-Hamiltonicityare pairwise disjoint we get that the size of every set E ⊆ { e ⊆ V ( G ) | | e | = 2 } , for which( V ( G ) , E ( G ) E ) is Hamiltonian, is at least λn/ (cid:15)dn and hence G is (cid:15) -far from beingHamiltonian. (cid:74) Note that for the caterpillar (see Figure 1) such pairwise disjoint sets of vertices, whoseneighbourhoods witness non-Hamiltonicity, are the singleton sets consisting of the vertices ofdegree one. Similarly, for the cycle of C ’s (see Figure 1), we can choose the vertex set ofevery other C on the cycle. On graphs of bounded degree that are far from being Hamiltonian v v v . . . v Figure 2 P ( v , . . . , v ). u u u u u u u w w w w w w v v v v v Figure 3
A link from P ( u , . . . , u ) to P ( v , . . . , v ) via w , . . . , w . One might wonder if the converse of Lemma 3 is true, i. e., if G is (cid:15) -far from beingHamiltonian, then G contains a linear fraction of pairwise disjoint sets of vertices whoseneighbourhoods witness non-Hamiltonicity. (This would actually imply the existence of aone-sided error property tester with constant query complexity.) Our examples below showthat this is not the case. In fact it even shows that for some constant c ∈ N there is a classof graphs which are (cid:15) -far from being Hamiltonian but we cannot even find c sets of verticeswith pairwise disjoint neighbourhoods witnessing non-Hamiltonicity. In other words there isa class of graphs which are (cid:15) -far from being Hamiltonian but δ -locally Hamiltonian for some δ, (cid:15) ∈ (0 , In this section we introduce the main step of our construction of graphs which are locallyHamiltonian and far from being Hamiltonian. At a high level, we construct a graph G E bychoosing a d -regular base graph E and building G E by introducing a path-gadget for everyedge of E , connecting these path-gadgets into a large cycle and linking path gadgets togetherif the edges of E corresponding to the path gadgets are incident to the same vertex. We givethe precise construction in the following.First we create a gadget (see Figure 2 for illustration). Let v , . . . , v be a set of vertices.Then we let P ( v , . . . , v ) be the graph with vertex set { v , . . . , v } and edge set (cid:8) { v i , v i +1 } , { v j , v j +3 } , { v k , v k +5 } | i ∈ { , . . . , } , j ∈ { , } , k ∈ { , , , } (cid:9) . For a graph G with { u , . . . , u , v , . . . , v , w , . . . , w } ⊆ V ( G ) and G [ u , . . . , u ] = P ( u , . . . , u ) and G [ v , . . . , v ] = P ( v , . . . , v ) we say that G contains a link from P ( u , . . . , u ) to P ( v , . . . , v ) via w , . . . , w (see Figure 3 for illustration), if E ( G ) contains n { u , v } , { v , u } , { u , v } , { v , u } , { u , w } , { w , w } , { w , w } , { w , v } , { v , w } , { w , w } , { w , w } , { w , u } o . Finally to any graph G we associate a directed graph ~G which is the graph that is obtainedfrom G by replacing every edge { u, v } ∈ E ( G ) by the two directed edges ( u, v ) and ( v, u ).We can now define the graph construction. . Adler and N. Köhler 7 (cid:73) Definition 4.
Let E be a d -regular graph (the base graph ) and f : E ( ~ E ) → { , . . . , | E ( ~ E ) |} be any linear order on E ( ~ E ) . We define the graph G E as follows. V ( G E ) := { a e , . . . , a e | e ∈ E ( ~ E ) } ∪ { b v , . . . , b v | v ∈ V ( ~ E ) } .E ( G E ) consists of the minimum set of edges such that G E [ a e , . . . , a e ] = P ( a e , . . . , a e ) for every e ∈ E ( ~ E ) , a f − ( i )31 is adjacent to a f − ( j )1 for every i ∈ [ | E ( ~ E ) | ] , j := i + 1 mod | E ( ~ E ) | and G E contains a link from P ( a ( v,w )1 , . . . , a ( v,w )31 ) to P ( a ( u,v )1 , . . . , a ( u,v )31 ) via b v , . . . , b v forevery triple of vertices u, v, w ∈ V ( ~ E ) with ( u, v ) , ( v, w ) ∈ E ( ~ E ) . See Figure 4 for an illustration. Note that the construction of G E depends on f as well as E ,but since the properties of G E are independent of which linear order f we use, we omit thedependency on f . (cid:73) Remark 5. If E is d -regular, for d ≥
1, and | V ( E ) | = n , then the degree of G E is at most d + 3 and | V ( G E ) | = (6 + 31 d ) n . (cid:73) Note. G E contains a large cycle of length 31 dn , i. e., the cycle( . . . . . . , a f − ( i − , a f − ( i )1 , a f − ( i )2 , . . . , a f − ( i )31 , a f − ( i +1)1 , . . . . . . ) . However G E also contains 6 n vertices which are not part of this cycle. In this section we prove the following. (cid:73)
Theorem 6.
For every d ∈ N > there is (cid:15) = (cid:15) ( d ) ∈ (0 , such that for any d -regular graph E the graph G E constructed in Definition 4 is (cid:15) -far from being Hamiltonian. To prove Theorem 6 we use the two technical Lemmas below (Lemma 8 and Lemma 9). Theywill be applied to graphs G obtained from G E by modifying a small fraction of the edgesof G E . Therefore they are stated for graphs G which share certain induced subgraphs with G E . The first of the two Lemmas (Lemma 8) states that if G has a Hamiltonian cycle and acertain induced subgraph, which also appears in G E , then the Hamiltonian cycle has certainsubpaths. The proof of Lemma 8 is illustrated in Figure 4. We will use the following easyobservation in the proof of Lemma 8. (cid:73) Remark 7.
Let G be a graph, u ∈ V ( G ) a vertex of degree 2 and v, w the two neighbours of u . Then any cycle C containing the vertices u, v and w must contain ( v, u, w ) as a subpath.Recall that subpaths appear either in the path or in the reversed path. (cid:73) Lemma 8.
Let E be any d -regular graph and G E as defined in Definition 4. Pick v ∈ V ( ~ E ) and let S v := { a ei | e ∈ E ( ~ E ) , e is incident to v } ∪ { b v , . . . , b v } . Let G be a graph with S v ⊆ V ( G ) . Assume G E [ N G E ( S v )] ∼ = G [ N G ( S v )] and f : S v → S v defined by f ( v ) = v for v ∈ S v is an isomorphism from G E [ S v ] to G [ S v ] . Then for every Hamiltonian cycle C in G and every edge e ∈ V ( ~ E ) incident to v the following properties hold. (i) Either ( a e , . . . , a e ) of C or ( a e , a e , a e , a e , a e ) is a subpath of C . (ii) Either ( a e , . . . , a e ) or ( a e , a e , a e , a e , a e ) is a subpath of C . (iii) Either ( a e , . . . , a e ) or ( a e , a e , a e , a e , a e , a e , a e , a e , a e ) is a subpath of C . (iv) If e ∈ E + G ( v ) then either ( a e , . . . , a e ) or ( a e , a e , a e , a e , a e , a e ) is a subpath of C . (v) If e ∈ E − G ( v ) then either ( a e , . . . , a e ) or ( a e , a e , a e , a e , a e , a e ) is a subpath of C . On graphs of bounded degree that are far from being Hamiltonian a ( v,w )1 a ( v,w )6 a ( v,w )12 a ( v,w )21 a ( v,w )27 a ( u,v )20 a ( u,v )26 a ( w,x )5 a ( w,x )11 b v b v b w b w Figure 4
Close-up of G E with vertices of high degree ( d + 1, d + 2 or d + 3) indicated by ‘fans’. Proof.
To prove (i) let us observe that a e and a e have degree 2 in G , as G E [ N G E ( S v )] ∼ = G [ N G ( S v )] and a e and a e have degree 2 in G E . Hence ( a e , a e ) and ( a e , a e , a e ) have to besubpaths of C as in Remark 7. Since a e has exactly three neighbours a e , a e and a e in G E and G E [ N G E ( S v )] ∼ = G [ N G ( S v )] we get that either ( a e , . . . , a e ) is a subpath of C or( a e , a e , a e , a e , a e ) is a subpath of C . Property (ii) follows with a similar argumentation.For (iii) let us assume that neither ( a e , . . . , a e ) nor ( a e , a e , a e , a e , a e , a e ) appearin C as a subpath. Since both a e and a e have degree 2 in G , we know that ( a e , a e , a e )and ( a e , a e , a e ) are subpaths of C . Hence neither ( a e , a e ) nor ( a e , a e ) are subpathsof C . Since both a e and a e have degree 3 in G , this implies that ( a e , a e , a e , a e , a e )is a subpath of C . Since a e has degree 2, then ( a e , a e , a e , a e , a e , a e , a e ) has to be asubpath of C . Since this is a cycle, C must be equal to ( a e , a e , a e , a e , a e , a e , a e ) whichcontradicts the assumption that S v is contained in C . A symmetric argument shows thateither ( a e , . . . , a e ) or ( a e , a e , a e , a e , a e , a e ) has to be a subpath of C , proving (iii).We will prove (iv) and (v) simultaneously using a counting argument. Let us first ob-serve that for every edge e ∈ E ( ~ E ) incident to v we know that ( a e , a e , a e ), ( a e , a e , a e ),( a e , a e , a e ) and ( a e , a e , a e ) are subpaths of C , because a e , a e , a e and a e have de-gree 2 in G . Let S be the set of all maximal subpaths of C which only contain ver-tices from { a e , . . . , a e , a ˜ e , . . . , a ˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } . Since there are no edgesof the form { a ei , a ˜ ej } for i, j ∈ { , . . . , , , . . . , } , e = ˜ e ∈ E ( ~ E ), every subpath in S is either of length 3 or length 6. For every path P = ( p , . . . , p ‘ ) ∈ S , we define thevertices u P , w P to be the neighbours of P on C , i.e. ( u P , p , . . . , p ‘ , w P ) is a subpath of C . Since G E [ N G E ( S v )] ∼ = G [ N G ( S v )] and every path P ∈ S is maximal, we know that u P , w P ∈ { a e , a e , a e , a e , a ˜ e , a ˜ e , a ˜ e , a ˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } ∪ { b v , b v } . Properties(i),(iii) imply that for every edge e ∈ E − ~ E ( v ), only one of the two vertices a e , a e and onlyone of the two vertices a e , a e can be in the set { u P , w P | P ∈ S } . Similarly, (ii),(iii)imply that for every edge e ∈ E + ~ E ( v ) only one of the two vertices a e , a e and only one of thetwo vertices a e , a e can be in the set { u P , w P | P ∈ S } . Furthermore there are two notnecessarily distinct edges e, ˜ e ∈ E + ~ E ( v ) such that ( a e , . . . , a e , b v ) and ( b v , a ˜ e , . . . , a ˜ e ) aresubpaths of C and hence the vertices a e , a e , a ˜ e , a ˜ e cannot be in { u P , w P | P ∈ S } . Hence |{ u P , w P | P ∈ S }| ≤ | E + ~ E ( v ) | − | E − ~ E ( v ) | + 2 = 4 d . In addition, note that (i),(ii), (iii)and deg G ( b v ) = 2 and deg G ( b v ) = 2 imply that no maximal subpath of C only containingvertices in S v \ { a e , . . . , a e , a ˜ e , . . . , a ˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } has length at most 1 andhence |{ u P , w P | P ∈ S }| = 2 | S | . Therefore | S | ≤ d . If any path in S has length 3 then | S | > d , since |{ a e , . . . , a e , a ˜ e , . . . , a ˜ e | e ∈ E − ~ E ( v ) , e ∈ E + ~ E ( v ) }| = 12 d . This yields acontradiction and hence (iv) and (v) are true. (cid:74) Let G be a graph with a ei , . . . , a ej ∈ V ( G ) for some edge e ∈ E ( ~ E ) and 1 ≤ i ≤ j ≤ C is a cycle in G which contains a ei , . . . , a ej . We say that C traverses the vertices . Adler and N. Köhler 9 a ei , . . . , a ej in order if ( a ei . . . , a ej ) is a subpath of C and we say that C traverses a ei , . . . , a ej out of order otherwise. Note that for certain 1 ≤ i ≤ j ≤
31 and e ∈ E ( ~ E ) there is only oneway in which a cycle C can traverse a ei , . . . , a ej out of order (as specified in Lemma 8).The next lemma shows that for every vertex v ∈ V ( ~ E ) and every Hamiltonian cycle C in G E the number of edges e ∈ E − ~ E ( v ) for which C traverses a e , . . . , a e out of order is exactlyone larger then the number of edges ˜ e ∈ E + ~ E ( v ) for which C traverses a ˜ e , . . . , a ˜ e out oforder. This still holds for every graph G which contains a certain induced subgraph of G E . (cid:73) Lemma 9.
Let E be any d -regular graph and G E as defined in Definition 4. Let S v := { a ei | e ∈ E ( ~ E ) , e is incident to v } ∪ { b v , . . . , b v } for some v ∈ V ( ~ E ) . Let G be a graph with S v ⊆ V ( G ) . Assume G E [ N G E ( S v )] ∼ = G [ N G ( S v )] and f : S v → S v defined by f ( v ) = v for v ∈ S v is an isomorphism from G E [ S v ] to G [ S v ] . Then for every Hamiltonian cycle C in G the cardinalities of the two sets T in v,C := n e ∈ E − ~ E ( v ) | ( a e , a e ) is a subpath of C o and (1) T out v,C := n e ∈ E + ~ E ( v ) | ( a e , a e ) or ( a e , b v ) is a subpath of C o (2) are equal. Proof.
Note that the condition G E [ N G E ( S v )] ∼ = G [ N G ( S v )] implies that no vertex in { a e , . . . , a e , a ˜ e , . . . , a ˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } ∪ { b v , . . . , b v } has neighbours in G \ S v .This will implicitly be used in the following argument whenever we exhaustively considerneighbours of vertices in G as successors on C .Let us first define a map F v,C : T in v,C → T out v,C , given by F v,C ( e ) := ˜ e , where ˜ e ∈ T out v,C is theedge such that ( a e , a ˜ e ) is a subpath of C . We first have to argue that F v,C is well defined.By Lemma 8 (iii), e ∈ T in v,C implies that ( a e , a e , a e , a e , a e , a e , a e , a e , a e ) is asubpath of C . Since the two neighbours a e and a e of a e are already part of this subpaththis implies that ( a e , a ˜ e ) has to be a subpath of C for some edge ˜ e ∈ E + ~ E ( v ). This implies that( a ˜ e , . . . , a ˜ e ) cannot be a subpath of C and hence, by Lemma 8 (iv), ( a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e )has to be a subpath of C . This further implies that ( a ˜ e , a ˜ e ) cannot be a subpath of C .Then if ( a ˜ e , . . . , a ˜ e ) is a subpath of C then ( a ˜ e , b v ) has to be a subpath of C by excludingall possible other neighbours of a ˜ e . On the other hand if ( a ˜ e , . . . , a ˜ e ) is not a subpathof C then, by Lemma 8 (iii), ( a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e ) is a subpath of C andhence ( a ˜ e , a ˜ e ) is a subpath of C . Therefore ˜ e ∈ T out v,C . This shows that F v,C is well defined.Furthermore F v,C is injective since if ( a e , a ˜ e ) and ( a e , a e ) are subpaths of C then ˜ e = e because ( a e , a e ) is also a subpath of C . F v,C is surjective as for ˜ e ∈ T out v,C both ( a ˜ e , a ˜ e )or ( a ˜ e , b v ) being a subpath of C together with Lemma 8 (iii) implies that ( a ˜ e , a ˜ e ) cannotbe a subpath of C . This further implies that ( a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e , a ˜ e ) is a subpath of C byLemma 8 (iv) and hence there is an edge e ∈ E − ~ E ( v ) such that ( a e , a ˜ e ) is a subpath of C .Then with the same argument as before ( a e , a e ) is a subpath of C and hence e ∈ T in v,C and F v,C ( e ) = ˜ e . Therefore F v,C is bijective which implies the statement of the lemma. (cid:74) As a direct consequence from Lemma 9 we get that G E cannot be Hamilonian for anybase graph E . That is true because if there is a Hamiltonian cycle C in G E then by Lemma 9the equation P v ∈ V ( ~ E ) | T in v,C | = P v ∈ V ( ~ E ) | T out v,C | must hold. But since every edge in T in v,C isalso contained in T out v,C and T out v,C must contain some edges (all the edges ( v, w ) for which( a ( v,w )12 , b v ) is a subpath of C ) that are not contained in T in v,C , the equation cannot hold andhence G E cannot be Hamiltonian. This argument works similarly if a small number of edgesin G E have been altered and the equality from Lemma 9 still has to hold for many vertices. Proof of Theorem 6.
Let (cid:15) := 1 / (8( d +3) (6+31 d )). Assume E is d -regular and n := | V ( G ) | .Let n := V ( G E ) = (6 + 31 d ) n and d := d + 3 the degree of G E .Towards a contradiction let us assume that G E is not (cid:15) -far to being Hamiltonian andlet E be a set of edges such that | E | ≤ (cid:15)d n and the graph G := ( V ( G E ) , E ( G E ) E ) isHamiltonian. Let B ⊆ V ( ~ E ) be the set of vertices defined by B := { v ∈ V ( ~ E ) | there is e ∈ E, i ∈ { , . . . , } , ˜ e ∈ E − G ( v ) ∪ E + G ( v ) such that a ˜ ei ∈ e }∪{ v ∈ V ( ~ E ) | there is e ∈ E, i ∈ { , . . . , } such that b vi ∈ e } . Note that | B | ≤ · (cid:15)d n , because every edge e ∈ E contributes at most 4 vertices to B , andhence | V ( ~ E ) \ B | ≥ n − (cid:15)d n > n/ C be a Hamiltonian cycle in G . Then for every vertex v ∈ V ( ~ E ) \ B we have that S v ⊆ V ( G ), G E [ N G E ( S v )] ∼ = G [ N G ( S v )] and f : S v → S v defined by f ( v ) = v for v ∈ S v is an isomorphism from G E [ S v ] to G [ S v ] where S v := { a ei | e ∈ E ( ~ E ) , e is incident to v } ∪{ b v , . . . , b v } . Since C is Hamiltonian C contains all vertices in S v for every v ∈ V ( ~ E ) \ B (amongst others). Hence by Lemma 9 we have | T in v,C | = | T out v,C | for every v ∈ V ( ~ E ) \ B where T in v,C and T out v,C are as define in Equation (1) and Equation (2). Therefore X v ∈ V ( ~ E ) \ B | T in v,C | = X v ∈ V ( ~ E ) \ B | T out v,C | . (3)As b v has precisely one neighbour in G for every v ∈ V ( ~ E ) \ B , which is not of the form a e for some e ∈ E + G ( v ) and this neighbour has degree 2 in G , we know that for precisely oneedge e ∈ E + G ( v ) the sequence ( b v , a e ) is a subpath of C . Hence X v ∈ V ( ~ E ) \ B (cid:12)(cid:12)(cid:12)n e ∈ E + G ( v ) | ( a e , b v ) is a subpath of C o(cid:12)(cid:12)(cid:12) = | E ( ~ E ) \ B | > n . (4)Since every edge ( u, v ) ∈ E ( ~ E ) such that u, v ∈ V ( ~ E ) \ B contributes 1 to both sides ofEquation (3), Equation (3) and Equation (4) imply that X v ∈ V ( ~ E ) \ B (cid:12)(cid:12)(cid:12)n ( u, v ) ∈ E ( ~ E ) | u ∈ B, ( a ( u,v )12 , a ( u,v )17 ) is a subpath of C o(cid:12)(cid:12)(cid:12) > n . But this is a contradiction as the number of edges ( u, v ) ∈ E ( ~ E ) for which u ∈ B is boundedfrom above by d | B | ≤ n/ (cid:74) In this Section we prove the following Theorem. (cid:73)
Theorem 10.
For any d -regular graph E with expansion ration h ( E ) ≥ the graph G E constructed in Definition 4 is δ -locally Hamiltonian for some constant δ = δ ( d ) ∈ (0 , . Our proof strategy for Theorem 10 is to add edges to G E which are incident to at most onevertex in N G E ( S ) to obtain a graph H which is Hamiltonian, for any given S ⊆ V ( G E ) ofsize at most δ | V ( G ) | . We prove the Hamiltonicity of H by dividing the vertex set of H intopairwise disjoint small sets. For each of these sets we obtain of a set of vertex disjoint pathswhich cover the entire small set and start and end in prescribed vertices. To conclude theprove of the Hamiltonicity of H we find a Hamiltonian cycle by patching together thesepaths. The next Lemma will be used to show the existence of such paths for all those subsetsof vertices of H which contain a vertex from S . . Adler and N. Köhler 11 (cid:73) Lemma 11.
Let E be any d -regular graph and G E as defined in Definition 4. Let v ∈ V ( ~ E ) and S v := { a e , . . . , a e , a ˜ e , . . . , a ˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } ∪ { b v , . . . , b v } . Let G be agraph such that G E [ S v ] is a subgraph of G . Then for any two sets T in v ⊆ E − ~ E ( v ) and T out v ⊆ E + ~ E ( v ) with | T in v | − | T out v | there is a set of d pairwise vertex disjoint simple paths { P in e , P out˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } in G with the following properties.If e ∈ T in v then P in e is a path from a e to a e .If e ∈ E − ~ E ( v ) \ T in v then P in e is a path from a e to a e .If e ∈ T out v then P out e is a path from a e to a e .If e ∈ E + ~ E ( v ) \ T out v then P out e is a path from a e to a e .The set { x ∈ V ( G ) | x is contained in P in e or P out e for some e } is equal to S v . ( ∗ ) Proof of Theorem 10.
Let δ := 1 / (2 · (6 + 31 d )) and let S ⊆ V ( G E ) be any set of verticeswith | S | ≤ δ · | V ( G E ) | . We will find a Hamiltonian graph H by modifying G E in such a waythat G E [ N G E ( S )] is not affected by any modifications.Let S v := { a e , . . . , a e , a ˜ e , . . . , a ˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } ∪ { b v , . . . , b v } for every v ∈ V ( ~ E ). Let S := { v ∈ V ( ~ E ) | S v ∩ S = ∅} . By Remark 5 | V ( G E ) | = (6 + 31 d ) · | V ( E ) | .Since the sets S v are pairwise disjoint this implies that | S | ≤ | S | ≤ δ · | V ( G E ) | = 1 / · | V ( E ) | .Let S = { s , . . . , s m } where m := | S | . (cid:66) Claim 12.
There are pairwise edge disjoint paths Q , . . . , Q m in E such that Q i is of theform Q i = ( q i , . . . , q ‘ i i ) for some ‘ i ∈ N and q ‘ i i = s i , q ji ∈ S for all j > q i ∈ V ( E ) \ S . Proof.
By induction on the size of S . If | S | = 1 then this is trivially true. If | S | = n then h ( E ) ≥ v with at least as many neighbours in V ( E ) \ S as neighbours in S . Then S \ { v } has n − S \ { v } . But then we can extend every path which starts in v by adifferent edge so it starts in V ( E ) \ S . (cid:67) Let Q , . . . , Q m be as in Claim 12. Further, for every vertex v ∈ V ( E ) \ S we pick one vertex u ∈ V ( E ) with ( v, u ) ∈ E ( ~ E ) and define n ( v ) := u . Now let E be the set (cid:26)n b v , a ( v,n ( v ))4 o , n b v , a ( v,n ( v ))13 o (cid:12)(cid:12)(cid:12)(cid:12) v ∈ V ( E ) \ S (cid:27) ∪ (cid:26)n a ( q i ,q i )14 , a ( q i ,q i )17 oo (cid:12)(cid:12)(cid:12)(cid:12) ≤ i ≤ m (cid:27) . We now define the graph H by setting V ( H ) := V ( G E ) and E ( H ) := E ( G E ) ∪ E . Note that H has degree d + 3, as we only added at most one edge to vertices of degree at most d + 1.Further note that by definition of S we have that S ⊆ S v ∈ S S v . Since every edge in E isincident to at most one vertex in N G ( S v ∈ S S v ) it follows that if H is Hamiltonian then itfulfils the conditions from Definition 2. Therefore, if we prove that H has a Hamiltoniancycle then G E must be locally Hamiltonian. (cid:66) Claim 13.
There is a set of 2 d pairwise vertex disjoint simple paths { P in e , P out˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } for every v ∈ V ( ~ E ) \ S with the following properties.If e ∈ E − ~ E ( v ) then P in e is a path from a e to a e .If e = ( q i , q i ) for some i ∈ [ m ] then P out e is a path from a e to a e .If e ∈ E + ~ E ( v ) \ { ( q i , q i ) | i ∈ [ m ] } then P out e is a path from a e to a e .The set { x ∈ V ( G ) | x is contained in P in e or P out e for some e } is equal to S v . ( ∗ )For v ∈ S we define the sets T in v := { ( q j − i , q ji ) | i ∈ [ m ] , j ∈ { , . . . , ‘ i } , q ji = v } and T out v := { ( q ji , q j +1 i ) | i ∈ [ m ] , j ∈ { , . . . , ‘ i − } , q ji = v } . Since for every v ∈ S there is exactly one path out of Q , . . . , Q m that ends in v , we get that | T in v | − | T out v | and hence the preconditions for Lemma 11 are met. Therefore we obtain a set of paths { P in e , P out˜ e | e ∈ E − ~ E ( v ) , ˜ e ∈ E + ~ E ( v ) } for every v ∈ S as in Lemma 11.Since S v ∩ S w = ∅ for every pair v, w ∈ V ( ~ E ) with v = w , we now have a set of pairwisevertex disjoint simple paths { P in e , P out e | e ∈ E ( ~ E ) } such that every vertex of H is containedin one of the paths. For every edge e ∈ E ( ~ E ) we now concatenate P out e with P in e to a path P e . This is possible as for every edge e ∈ E ( ~ E ) the end vertex of P out e and the start vertexof P in e are adjacent. Finally we concatenate all paths P e in the order given by the ordering f : E ( ~ E ) → [ | E ( ~ E ) | ] used in the construction of G E . This gives us a cycle which containsevery vertex in H precisely once. Hence H is Hamiltonian. (cid:74)(cid:73) Theorem 14.
There are d ∈ N and constants δ := δ ( d ) , (cid:15) := (cid:15) ( d ) ∈ (0 , and a sequence of d -bounded degree graphs ( G N ) N ∈ N of increasing order such that G N is δ -locally Hamiltonianand (cid:15) -far from being Hamiltonian for every N ∈ N . ( ∗ ) In this section we will introduce the bounded-degree model of property testing as introducedin [8] and then use our main result from the previous section to prove a lower bound for thecomplexity of property testing Hamiltonicity.Let d ∈ N and let C d be the class of graphs of bounded degree d . From now on, allgraphs have d -bounded degree. A property P on C d is any subset of C d which is closed underisomorphism. An algorithm that processes a graph G does not obtain an encoding of G as abit string in the usual way. Instead, it has direct access to G using an oracle which answersneighbour queries in G in constant time. In addition, the algorithm receives the number n ofvertices of G . We assume that the vertices of G are numbered 1 , , . . . , n . The oracle acceptsqueries of the form ( i, j ), for i ≤ n , and j ≤ d , to which it responds with the j -th neighbourof i , or with ⊥ if i has less than j neighbours.The running time of the algorithm is defined as usual, i. e. with respect to n . We assumea uniform cost model, i. e., we assume that all basic arithmetic operations including randomsampling can be performed in constant time, regardless of the size of the numbers involved. Distance.
For two graphs G and H , both with n vertices, dist( G, H ) denotes theminimum number of edges that have to be modified (i. e. inserted or removed) in G and H tomake G and H isomorphic. for (cid:15) ∈ [0 , G and H are (cid:15) - close if dist( G, H ) ≤ (cid:15)dn . If G, H are not (cid:15) - close , then they are (cid:15) -far . Note that in particular, G and H are (cid:15) -far if theirvertex numbers differ. A graph G is (cid:15) -close to a property P if G is (cid:15) -close to some H ∈ P .Otherwise, G is (cid:15) -far from P . Note that this generalises Definition 1. (cid:73) Definition 15 ( (cid:15) -tester) . Let
P ⊆ C d be a property and (cid:15) ∈ (0 , . An (cid:15) - tester for P is aprobabilistic algorithm with oracle access to an input G ∈ C d and auxiliary input n := | V ( G ) | .The algorithm does the following. If G ∈ P , then the (cid:15) -tester accepts with probability at least / . If G is (cid:15) -far from P , then the (cid:15) -tester rejects with probability at least / .An (cid:15) -tester is called a one-sided error tester if it accept every graph G ∈ P with probability . The query complexity of an (cid:15) -tester is the maximum number of oracle queries made withrespect to n . Let f : N → R be a function. A property P is testable with (one-sided errorand) query complexity f ( n ), if for each (cid:15) ∈ (0 ,
1] and each n , there is a (one-sided error) (cid:15) -tester for P ∩ { G ∈ C d | | V ( G ) | = n } on inputs from { G ∈ C d | | V ( G ) | = n } with querycomplexity f ( n ).We now obtain the following result as a corollary of Theorem 14. . Adler and N. Köhler 13 (cid:73) Theorem 16.
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Lemma 11.
First we pick a vertex n ( v ) ∈ V ( ~ E ) such that ( v, n ( v )) / ∈ T out v . This is possiblebecause v has the same number of incoming and outgoing edges and | T in v | − | T out v | . Then | T in v | = | T out v ∪ { ( v, n ( v )) }| , and hence we can find a bijection g : T in v → T out v ∪ { ( v, n ( v )) } .Then we can define the paths as follows. For e ∈ T in v we let P in e := ( a e , a e , a e , a g ( e )8 , a g ( e )7 , a g ( e )6 , a g ( e )11 , a g ( e )10 , a g ( e )9 , a e , a e , a e , a e , a e ) ,P out g ( e ) := ( a g ( e )1 , . . . , a g ( e )5 , b v , b v , b v , a e , a e , a e , a e , a e , a e , b v , b v , b v , a g ( e )12 , . . . , a g ( e )17 )if g ( e ) = ( v, n ( v )) and P out g ( e ) := ( a g ( e )1 , a g ( e )2 , a g ( e )5 , a g ( e )4 , a g ( e )3 , a e , a e , a e , a e , a e , a e , a g ( e )14 ,a g ( e )13 , a g ( e )12 , a g ( e )17 ,a g ( e )16 , a g ( e )15 )if g ( e ) = ( v, n ( v )).Furthermore for e ∈ E − ~ E ( v ) \ T in v we let P in e := ( a e , . . . , a e ) and for e ∈ { ( v, w ) ∈ E + ~ E ( v ) \ T out v we let P out e := ( a e , . . . , a e ). These paths clearly satisfy all conditions. (cid:74) Proof of Claim 13.
This can be achieved by letting P in e := ( a e , . . . , a e ) for e ∈ E − ~ E ( v ).Additionally, for every edge e = ( q i , q i ) we let P out e := ( a e , . . . , a e , a e , a e , a e ) if q i = n ( q i ) and P out e = ( a e , . . . , a e , b v , b v , b v , a e , . . . , a e , b v , b v , b v , a e , a e , a e , a e , a e ) otherwise.Finally for e ∈ E + ~ E ( v ) \{ ( q i , q i ) | i ∈ [ m ] } we set P out e := ( a e , . . . , a e ) for e = ( v, w ), w = n ( v )and P out e := ( a e , . . . , a e , b v , b v , b v , a e , . . . , a e , b v , b v , b v , a e , . . . , a e ) for e = ( v, n ( v )). (cid:67) Proof of Theorem 14.
Let D ∈ N and ( E N ) N ∈ N a sequence of D -bounded degree expandersof increasing order. Such expanders exists and there are even some known explicit construc-tions see for example [15] or [18]. Then for every N ∈ N we set G N := G E N be the graph . Adler and N. Köhler 15 constructed in Definition 4. By Theorem 6 and Theorem 10 there is a degree bound d andconstants δ, (cid:15) ∈ (0 , D , such that G N has degree bounded by d and G N is δ -locally Hamiltonian and (cid:15) -far from being Hamiltonian. (cid:74) B Proofs of Section 6
Proof of Theorem 16.
Pick d as in Theorem 14 and let P ⊆ C d be the class of all Hamiltoniangraphs of degree at most d . Towards a contradiction, assume that for every (cid:15) ∈ (0 ,
1] and n ∈ N there is a one sided-error (cid:15) -tester for P ∩ { G ∈ C d | | V ( G ) | = n } with query complexity o ( n ). Let δ, (cid:15) ∈ (0 ,
1) be constants such that there is a sequence of d -bounded degree graphs( G N ) N ∈ N of increasing order such that G N is δ -locally Hamiltonian and (cid:15) -far from beingHamiltonian for every N ∈ N . Note that δ and (cid:15) exist by Theorem 14. Let T be an (cid:15) -testerfor P with query complexity f ( n ) ∈ o ( n ). Since f ( n ) ∈ o ( n ) there must be n ∈ N suchthat f ( n ) ≤ δn for all n ≥ n . Let N ∈ N such that | V ( G N ) | ≥ n . Since G N is (cid:15) -far from P there must be a sequence of queries ( q , . . . , q m ) with m ≤ δn such that T queries thesequence ( q , . . . , q m ) with non-zero probability and rejects G N with non-zero probabilityafter performing the queries ( q , . . . , q m ). Let S be the set of vertices v ∈ V ( G N ) such thatthere is a query q i = ( v, j ) for i ∈ [ m ]. Because G N is δ -locally Hamiltonian and | S | ≤ δn there is a graph H ∈ P on n vertices and T ⊆ V ( H ) such that there is an isomorphism G N [ N G N ( S )] to H [ N H ( T )] which maps S to T . Hence, after renaming the vertices in N H ( T )),the tester T gets exactly the same answers for queries in q , . . . , q m for G N and H . Thisimplies that T queries the sequence ( q , . . . , q m ) in H with non-zero probability and hencemust rejects H with non-zero probability. This contradicts the assumption that T was aone-sided error tester for Hamiltonicity. (cid:74)(cid:73) Note 17.
Note that the above argument is not sufficient for two-sided error testers, becausea two-sided error tester would be allowed to reject H with probability < //