4.5 Trees with many leaves
4.5.2 Proof of Lemma 4.23
In this section we describe how to modify the embedding algorithm of Section 4.3.4 so that it successfully embeds a spanning tree T with many vertex-disjoint leaf- edges to a digraph G of high semidegree (thus proving Lemma 4.23).
Lemma 4.23. Suppose that 1n λ α and that 1n C1. If T is an oriented tree of order n with ∆(T ) ≤ (log n)C and at least λn vertex-disjoint leaf-edges, and if
G is a digraph of order n with minimum semidegree (12 + α)n then G contains a (spanning) copy of T .
Proof. As in the proof of Lemma 4.20, we begin by constructing a suitable regular partition of G. We introduce constants ε, d, η with 1/n 1/k ε d λ η α and apply Lemma 4.2 to obtain a partition V0 ˙∪ V1 ˙∪ · · · ˙∪ Vk and a digraph R? with V (R?) = V
0 ˙∪ [k] such that
(a) |V0| < εn and m := |V1|= · · · = |Vk|;
(b) For each i ∈ [k] we have G[Vi−1→Vi] and G[Vi→Vi+1] are (d, ε)-super-regular; (c) For all i, j ∈ [k] we have i→j ∈ E(R?) precisely when G[V
i→Vj] is (d, ε)- regular.
(d) For all v ∈ V0 and all i ∈ [k] we have v←i ∈ E(R?) precisely when
deg−(v, V
i) ≥ (1/2 + η)m, and v→i ∈ E(R?) precisely when deg+(v, Vi) ≥ (1/2 + η)m ;
(e) For all i ∈ [k] we have deg0
R?(i, [k]) ≥ (1/2 + η)k; and (f) For all v ∈ V0 we have deg0R?(v, [k]) > αk.
Let H ⊆ R? be the directed cycle 1→2→ · · · →k→1. Note that H, T and R? satisfy the conditions of Lemma 4.22 (applied taking the value of η for α there, with remaining constants as here), so there exists an allocation ϕ of the vertices of T such that
(i) Either all edges in E are oriented towards the leaf vertex, or all edges in E are oriented away from the leaf vertex.
4.5. Trees with many leaves 115 (iii) ϕ maps at least λn/32k leaf edges in E to each edge of H; and
(iv) ϕ −1(1) = ϕ −1(2) = · · · = ϕ −1(k) .
We assume that all edges in E are oriented towards the leaf vertex; the proof is similar otherwise. Let L0 := ϕ−1(V0) be the set of leaves which are mapped to V0,
and let P0 be the set of parents of those leaves, so |V
0|= |L0|. For each i ∈ [k],
let Li be a set containing precisely λm/32 leaves mapped to Vi whose parents have been mapped to Vi−1, let Pi be the set of parents of Li, and let LH =Si∈[k]Li.
We embed T0 := T \ LH to G by applying a (slightly modified) version of the vertex embedding algorithm. Before doing so, we reserve some sets of vertices of G which have good properties and which we will use to complete the embedding later on. We introduce a new constant γ with 1/n 1/k ε γ d. For each i ∈ [k] reserve sets Xi (for the final matching) and Yi for parents of whose leaves will be embedded to V0.
Claim 4.24. There exist v1 ∈ V1 and, for each i ∈ [k], disjoint sets Xi, Yi ⊆ Vi with |Xi|= |Yi|= λm/100 such that
(i) If U ⊆ Vi with |U| ≥ λm/24, then G[Xi−1→U] and G[U→Xi+1] are both
100ε λ , d 32 ! -super-regular;
(ii) For all x ∈ ϕ−1(V0), if ϕ maps an inneighbour of x to V
j, then deg−
ϕ(x), Yj
≥ λm/400 ; (iii) For all y ∈ ϕ−1(V0), if ϕ maps an outneighbour of x to V
j, then deg+
ϕ(y), Yj
≥ λm/400 ; (iv) For each y ∈ C−(t1) and each z ∈ C+(t1) we have
deg−(v
1, Vϕ(y)) ≥ γm and deg+(v1, Vϕ(z)) ≥ γm.
Proof. This Claim (and its proof) are almost identical to Claim 4.21, so the proof
is omitted.
Returning to the proof of the lemma, (as in the proof of Lemma 4.20) we introduce a constant β (as above) with 1/n 1/k ε γ β, d. We next apply the embedding algorithm to T0 := T \ LH to allocate T0 to G \S
i∈[k]Xi (being careful when close to V0 to embed in the sets Yi all parents of leaves mapped to V0). Roughly speaking, as in Lemma 4.20, we embed each vertex of
116 Chapter 4. Spanning Structures via Semidegree T0 to G0 := G \S
i∈[k]Xi according to the allocation ϕ as dictated by the vertex embedding algorithm, except for the leaves in L0 and their parents in P0, which
we embed to V0 andS
i∈[k]Yi, respectively. More precisely, we apply the following changes to the embedding algorithm.
Step 1. For each i ∈ [k] write Yiτ for the available vertices of Yi, write Viτ for the available vertices in Vi\(Xi ˙∪ Yi) and change the definition of Bτ so that it now includes Yτ
1 ∪ · · · ∪ Ykτ, i.e. let Bτ := {v1, . . . , vτ−1} ∪ Y1τ∪ · · · ∪ Ykτ ∪ [ ts: ts is open A−s ∪ A+s, so for all τ ≥ 1 and all i ∈ [k] we have Vτ
i ∩ Yi = ∅.
Step 2. Nothing changes in this step.
Step 3. We only modify this step if tτ is either a vertex in P0 or a parent of such vertex, otherwise we proceed as in the original algorithm.
If in Step 2 we embedded tτ ∈ P0 to a vertex vτ, then tτ is adjacent to a leaf ` ∈ L0 with w` := ϕ(`) ∈ V
0; moreover, tτ was embedded to an inneighbour of w` in Yτ
ϕ(tτ). We reserve a set A
+
` := {w`} for the child of tτ, and let A−τ and A+τ be the union of the sets reserved for the other children of tτ, which we select as in Step 3 of the original algorithm. If in Step 2 we embedded a vertex tτ which is a parent of a vertex
p ∈ P0, we reserve sets for the other children of tτ, as in the original algorithm, but reserve the set A+
p (or A−p) for p in a different manner, so that it is guaranteed to lie in Yτ
ϕ(p)∩ N−
ϕ(`), where ` is the only leaf in L0 connected to p in T : if p ∈ C−(t
τ), choose a set A−tτ ⊆
NG−(vτ) ∩ Yϕτ(p)∩ N−
ϕ(`)containing at most 2m3/4 vertices and which is (β, γ, ϕ, m)-good for Sp; and if p ∈ C+(tτ), choose a set A+tτ ⊆
NG+(vτ) ∩ Yϕτ(p)∩ N−
ϕ(`)containing at most 2m3/4 vertices and which is (β, γ, ϕ, m)-good for Sp. (In either case we let A−τ and A+τ be the union of the sets reserved for the children of tτ, as in the original algorithm.) If neither of these conditions apply, we follow Step 3 of the original algorithm.
We now argue that the modified embedding algorithm successfully embeds T0 to G \ (X1∪ · · · ∪ X
k) and that every vertex of P0 is embedded to a vertex in Y1 ˙∪ · · · ˙∪ Yk. Note first that every vertex is embedded according to ϕ (i.e., for all x ∈ T we embed x to ϕ(x) if ϕ(x) ∈ V0, and embed x to a vertex in Vϕ(x) otherwise). Following the proof of Lemma 4.16 all that we need to prove is that
4.5. Trees with many leaves 117 the choices of the embedding algorithm can be carried out as required. Note that if the choices in Step 3 can be done, then all choices in Step 2 (all of which are made according to the original algorithm) can be made; as a consequence, we only need to consider what happens in Step 3. Recall that T0 has none of the
leaves in LH and that for each i ∈ [k] we have |ϕ(LH) ∩ V
i|= |LH|/k; therefore, for all τ ≥ 1 and all i ∈ [k] we have |Vτ
i | ≥ |Li| − |Xi| − |Yi| ≥ λm/50; moreover, for each p ∈ P0 connected to a leaf ` ∈ L0, the number of vertices in reserved
sets at any time τ is at most 2m3/4(log n)∆(T ) ≤ εm ≤ |Yτ
ϕ(p) ∩ N−
ϕ(`)|/2. When the algorithm reaches Step 3, we have just embedded a vertex tτ ∈ T0 to a vertex vτ ∈ G \Si∈[k]Xi. We consider the following 3 cases.
If tτ ∈ P0, then tτ is adjacent to a single leaf ` ∈ L0, and this leaf is mapped to a vertex w` := ϕ(`) ∈ V
0. Since ϕ−1(w`) = {`} and vτ→w` ∈ E(G) we can reserve the desired set A+
` . Moreover, since |Viτ| ≥ |Li| − |Xi| − |Yi| ≥ λm/50 we can apply Lemma 4.4 to find and reserve good sets for each of the remaining children of tτ.
Now suppose tτ is a parent of a vertex p ∈ P0. As before, p is adjacent to a single leaf ` ∈ L0, and this leaf is mapped to a vertex w` := ϕ(`) ∈ V
0.
We wish to reserve sets for the children of tτ with the restriction that the set reserved for p should lie in Yτ
i ∩ NG−(w`). Recall that the only vertices we ever embed to Yi lie in P0, so the number of vertices of Yi unavailable for embedding is at most |P0| + 2m3/4(log
2n)∆(T ) ≤ 2εm. By (ii), it follows
that |Yτ
i ∩ NG−(w`)| ≥ λm/400 − 2εm ≥ γm, so we can reserve a set for p which is good for Sp as required as well.
Lastly, if neither of the previous conditions holds, then we reserve sets for the children of tτ as in the original algorithm; this can be done since
|Viτ| ≥ |Li| − |Xi| − |Yi| ≥ λm/50.
Since it is always possible to reserve the desired sets, we conclude that the modified embedding algorithms successfully embeds T0 to G \S
i∈[k]Xi.
Let Pi−1 be the parents of Li, so |Pi|= |Li|and, every vertex in Pi−1 has been
embedded to i − 1. For each i ∈ [k], let Wi be the set of vertices of Vi ⊆ G to which no vertex has been embedded yet, and let Ui be the set of vertices to which the vertices in Pi have been embedded. Since |Wi| = |Li|= |Ui−1| and Xi ⊆ Li we have that there exists a perfect matching of edges directed from Ui−1 to Wi by Claim cl:reserve-again (i) and Lemma 2.8. This completes the embedding of T to G.
118 Chapter 4. Spanning Structures via Semidegree