Near-decompositions into spanning rainbow trees

Now we combine everything from this section to prove the asymptotic version of the Brualdi-Hollingsworth and Kaneko-Kano-Suzuki Conjectures. We will need the following standard lemma.

Lemma 9.10. Every graphGwith e(G)≥(1−(ε/2)2_)n2_{/2 has an induced subgraphH with δ(H)≥(1−ε)n.}

Proof. LetS be the set of verticesv in Gwith d(v)≤(1−ε/2)n. We have 2e(G)≤(n− |S|)n+|S|(1−ε/2)n

which combined with e(G) ≥ (1−(ε/2)2_)n2_{/2 gives |S| ≤ εn/2. Let H = G\S to get a graph with δ(H) ≥}

We will also need the following lemma about switching edges between a tree and a forest.

Lemma 9.11. Let T be a tree and F a forest all of whose edges touch V(T). Then, there is a tree T0 which containsF and is contained inT ∪F.

Proof. Notice thatT∪F is connected sinceT is a tree and all edges ofF touchT. LetT0be a connected subgraph of T∪F which contains F and has e(T0) as small as possible. If T0 is acyclic then we are done. Otherwise, T0

contains a cycleC. SinceF is a forest C must contain at least one edge ofT. Deleting this edge gives a smaller connected graph contradicting the minimality ofe(T0).

By combining everything in this section with our earlier Hamiltonian decompositions we can show that the Brualdi-Hollingsworth and Kaneko-Kano-Suzuki Conjectures hold asymptotically when the colouring on Kn is

close to a 1-factorization.

Lemma 9.12. Let 1 poly ε,log−1n poly γ poly n−1_{. Let K}

n be properly coloured with ≥ (1−γ)n colours having

≥(1−γ)n/2 edges. Then, Kn has(1−8ε)n/2 edge-disjoint spanning rainbow trees.

Proof. Choose 1polyε,log−1npolyηpolyβ,ˆk−1poly_ν poly_γ 1

poly

γpolyn−1_.

Set aside small colours: LetC be the set of colours with≥(1−γ)n/2 edges inKn. By the assumption of

the lemma andγ1

poly

γwe have eKn(C)≥(1−γ)

2_n2_{/2≥(1−(γ}

1/2)2)n2/2.

By Lemma 9.10 applied toKn[C] withε=γ1there is a subgraphGofKn withδ(G)≥(1−γ1)n, having only

colours ofC. Setn1=|G| ≥(1−γ1)nand notice that Gis (3γ1,1, n1)-typical.

Choose set S: Fixn2=d(1−ν)ne. Apply Lemma 5.2 (c) withp=n2/n1,n0 =n1, µ= 1/2, and γ0 = 3γ1

in order to find a set of verticesS⊆V(G) of ordern2 withG[S] globally (1 + 3γ1)(n2/2n1)n2-bounded, andG[S]

(6γ1,1, n2)-typical. Notice that G[S] is globally (1−0.9ν)n2/2-bounded (using n1≥(1−γ1)n,n2 =d(1−ν)ne

and 1polyν polyγ1). Notice that inG[S] any colour ofCcovers at least≥(1−γ)n−(n−n2)≥(1−2ν)n2vertices.

Set aside a pseudorandom graph H: PartitionG[S] into subgraphsG1 and H with every edge placed in

H independently with probabilityη. By Lemma 5.2 (b),G1 is (12γ1,1−η, n2)-typical and globally (1 + 6γ1)(1−

η)(1−0.9ν)n2/2-bounded (for the application takep= 1−η, µ= (1−0.9ν)/2,n0=n2, δ= 1,γ0 = 6γ1). Since

νpolyγ1,G1is globally (1−0.5ν)(1−η)n2/2-bounded. By Lemma 5.4 applied withp=η,ε0 =ε,k0= ˆk,ν0 = 2ν,

H has the property that any set of ˆkcolours ofC cover≥(1−8ν)n≥(1−ε)nvertices.

Find near-decomposition ofKn[S]into rainbow paths: Apply Lemma 8.27 toG1withn0 =n2,γ0= 12γ1,

p= 0.5ν, δ = 1−η in order to find (1−0.5ν)(1−η)n2 edge-disjoint rainbow Hamiltonian paths in G1. Using

(1−0.5ν)(1−η)n2≥(1−ε)n/2 choose a subcollectionP1, . . . , Pb(1−ε)n/2c of these paths. SinceG1is a subgraph

ofG, these paths only use edges with colour inC.

Add small colours into trees: Let CL be the set of colours with ≥ (1−ε)n/2 edges in Kn. Choose

k = max(n−1− |CL|,0). By assumption we have k ≤γn. LetG2 be the subgraph of Kn consisting of edges

with colour outside CL which touch S. We claim that e(G2) ≥ (1 +η)kb(1−ε)n/2c. When k = 0, this is

obvious. Otherwise since δ(Kn) =n−1 andKn is properly coloured, we havee(G2)≥ 1₂P_v∈SdC(Kn)\CL(v)≥

|S|(δ(Kn)−|CL|)/2 =kd(1−ν)ne/2≥(1+η)k(1−ε)n/2. By definition ofCL, the graphG2is globallyb(1−ε)n/2c-

bounded. Apply Lemma 9.5 toG2 withm=b(1−ε)n/2c,ε0 = 1.01ν, β0 =η,S=S. This gives us edge-disjoint

rainbow forestsF1, . . . , Fb(1−ε)n/2c of sizekin G2.

Apply Lemma 9.11 fori= 1, . . . ,b(1−ε)n/2ctoPiandFi in order to find a rainbow treeTicontainingFiand

contained inPi∪Fi(Tiis rainbow sincePiandFiare colour-disjoint which happens becauseC(Pi)⊆C⊆CLand

C(Fi)∩CL=∅). In particular, eachTi containskedges outsideCL (the edges ofFi). Sincek≥n−1− |CL|, this

implies that eachTi avoids k+|CL| −e(Ti)≥n−1−e(Ti) colours of CL, each of which has ≥(1−ε)n/2 edges

inKn. Additionally, from Lemma 9.5, we have that for every vertexv6∈S eitherv∈Ti for alliordTi(v)≤1 for

alli. LetS0 =S∪ {v6∈S:v∈Ti for eachi}and notice that|S0| ≥ |S|=d(1−ν)ne. Now for eachi andv6∈S0,

we havedTi(v)≤1 and alsoS

0_⊆V(T

i).

Make trees spanning: Observe thatH is disjoint fromG1andG2. (The former holds by construction ofG1.

The latter byC(H)⊆C⊆CL andC(G2)∩CL =∅), and henceH is disjoint from the trees T1, . . . , Tb(1−ε)n/2c.

Apply Lemma 9.8 with S = S0, trees T1, . . . , T(1−8ε)n/2, H = H, β0 = ε, k0 = ˆk and ε0 = ν in order to find

(1−8ε)n/2 edge-disjoint spanning rainbow trees inKn, where we have used thatε,1/ˆk

poly ν.

Combining the above with our earlier result about Hamiltonian decompositions, we prove that the Brualdi- Hollingsworth and Kaneko-Kano-Suzuki Conjectures hold asymptotically.

Theorem 9.13. Let 1 poly εpoly n−1. Every properly colouredKn has (1−ε)n/2 edge-disjoint spanning rainbow

trees.

Proof. Fix 1polyε,log−1npolyγpolyn−1_{. IfK}

n has≥(1−γ)ncolours having≥(1−γ)n/2 edges, then the theorem

follows from Lemma 9.12. Otherwise, Kn has ≤(1−γ)n colours having≥ (1−γ)n/2 edges, and the theorem

follows from Lemma 8.29.