Justificación de los estudios de imitación

LetT be a tree on n vertices, rooted at t1, letH ⊆ V (T ), and let k be a positive integer. For any vertexx _{∈ T , there is a unique path in T from x to t}1; letPx denote the set of the firstk vertices of this path, starting fromx. Let H1 ₌S

x∈HPx, and then for eachi > 1 let Hi+1be formed fromHi _{by adding the vertices ofP}

x for anyx ∈ Hi with at least two children in Hi. After at mostn steps we must have Hi _{= H}i+1_{, when we terminate the process. We refer to} this finalHi_{asH with leading paths included, denotedP}

k(H). So H ⊆ Pk(H)⊆ V (T ). Note that_Pk(H) depends on both the value of k and the root t1ofT .

Next we prove two results which enable us to make use of this definition. The first shows that ifH is small thenPk(H) is small, and the second shows that if H is small then it is possible to embed any componentT0ofT [Pk(H)] in a regular and dense cycle of cluster tournaments such that the vertices ofV (T0)∩ H are embedded in the first cluster and the ‘root’ of H is embedded

in a given cluster.

Proposition 2.23 Letk be any positive integer, let T be a tree on n vertices, rooted at some

t1 ∈ T, and letH⊆ V (T ). Then|Pk(H)| 6 3k|H|.

Proof. Consider any componentT0 of T [Pk(H)], and let t01 be the unique vertex of T0 with minimald(t1, t01). Then every vertex of T0 lies on the path from some vertex ofH to t1, and so T0 _{is precisely the set of vertices in paths between t}0

1 and vertices of H ∩ V (T0). Thus onlyt0

1 and vertices ofH can be leaves of T0. It follows thatT [Pk(H)] has at most 2|H| leaves. Since T [_Pk(H)] is a forest, it follows that the number of vertices of T [Pk(H)] with at least two children inT [_Pk(H)] is also at most 2|H|. Furthermore, any vertex x ∈ T for which the vertices ofPx were added to Pk(H) at any stage is either a member of H or has at least two children in_Pk(H). This is true for at most 3|H| vertices x, and for each such vertex at most k

Lemma 2.24 Suppose that 1/m  1/k  ε  d. LetT be a directed tree rooted at some

t1 ∈ T. LetH ⊆ V (T )be of size |H| 6 m/10k, letT0 be a component ofT [Pk(H)]which

does not containt1, and lett01 be the unique vertex ofT0 with minimald(t01, t1). Let Gbe an

ε-regulard-dense cycle of cluster tournaments on clustersV1, . . . , Vk, each of sizem. Then for

anyj _{∈ [k]},Gcontains a copy ofT0 _{with the vertext}0

1 corresponding to some vertex ofVj, and

every vertex inV (T0_{)∩ Hcorresponding to some vertex ofV} 1.

Proof. Informally, from the perspective of t0

1, T0 begins with a path of lengthk − 1 (from t01 tot, say) before possibly branching out. So we find a copy of T0 _{inG by first embedding the} vertices of this path so thatt0

1is embedded inVj andt is embedded in V1. We then embed all of the remaining vertices ofT0 _inV

1.

More formally, note that for each 0 6 s 6 k − 1 there is precisely one vertex xs of T0 with

d(t01, xs) = s (so x0 = t01, and xi ∈ H for any i < k − 1). Let F ⊆ [k − 1] be the set of/ thoses such that x_{s−1 → x}s, and let B ⊆ [k − 1] be the set of s such that xs−1 ← xs. Then

|F | + |B| = k − 1, so either |F | > k − j or |B| > j − 1. Suppose first that |B| > j − 1. Then

chooseB0 _{⊆ B of size j − 1. We allocate the vertices of T}0 _{to the clustersV}

1, . . . , Vj. Begin by allocatingx0 toVj. Then for eachs ∈ [k − 1] in turn, let Vi be the cluster to which xs−1 was allocated, and allocatexstoViifs /∈ B0, or toVi−1ifs∈ B0. Then since|B0| = j − 1, xk−1 is assigned toV1. Finally, allocate all other vertices ofT0toV1. Then every edge ofT0is allocated either canonically or within a cluster.

Next we embed T0 _{in G so that every vertex is embedded within the cluster to which it is} allocated. To begin, by a standard regularity argument we may choose for eachi a set V0

i ⊆ Vi so that _|Vi0_{| > 9m/10 and every vertex v ∈ Vi}0 has at least dm/2 outneighbours in V0

i+1. Let G0 _{= G[V}0

1 ∪ · · · ∪ Vk0]. Now, for each i, let Si be the set of vertices of T0 allocated to

Vi. So |S2|, . . . , |Sk| 6 k − 1 and |S1| 6 |T0|. Then by Proposition 2.23, 3|S1| 6 3|T0| 6

9k|H| 6 |V0

have successfully embeddedT0[S1 ∪ · · · ∪ Si−1] in G[V10 ∪ · · · ∪ Vi0−1] for some i 6 j. Then precisely one vertext _{∈ S}i has a neighbour t0 ∈ Si−1, and t0 has already been embedded to somev0 _{∈ V}0

i−1. Nowv0has at leastdm/2 > 3|Si| outneighbours in Vi0, and so by Theorem 1.2 we may embedT0_[S

i] among these outneighbours. Let v be the vertex to which t is embedded; then since v is an outneighbour of v0_{, we have extended our embedding to an embedding of}

T0_[S

1∪ · · · ∪ Si] in G[V10∪ · · · ∪ Vi0]. Continuing in this manner we obtain an embedding of T0 inG, with t0₁ embedded inVj andV (T0)\ {x0, . . . , xk−2} ⊇ V (T0)∩ H embedded into V1, as desired. A similar argument achieves this if_{|F | > k − j.}