Fractional decompositions Fractional decompositions of dense hypergraphs of dense hypergraphs Raphael Yuster Raphael Yuster University of Haifa University of Haifa
Fractional decompositions of Fractional decompositions of dense hypergraphsdense hypergraphs
Raphael YusterRaphael YusterUniversity of HaifaUniversity of Haifa
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Definitions, notations and background
The Rödl nibble
The edges of the complete r-graph K(n,r) can be packed, almost completely, with copies of K(k,r), where k is fixed.
This result is considered one of the most fruitful applications of the probabilistic method.
It was not known whether the same result also holds in a dense hypergraph setting.
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Example: Packing 7 copies of K(3,2) in a K(7,2)
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Definitions, notations and background
Let H0 be a fixed hypergraph.A fractional H0-decomposition of a hypergraph H is an assignment of nonnegative real weights to the copies of H0 in H such that for each e E(H) the sum of the weights of copies of H0 containing e is 1.
Example:K4 has a fractional K3-decomposition.Each triangle receives a weight of ½.
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Main result
There exists a positive constant α=α(k,r) so that everyn-vertex r-graph in which every (r-1)-set is contained in at least (n-r+1)(1-α) edges has a fractionalK(k,r)-decomposition.
• The proof is algorithmic.
• K(n,r) is simply the case α=0 since every r-1 set is contained in n-r+1 edges.
• In fact we obtain: α(k,r) > 6-kr
α(k,2) > 0.1k -10
α(3,2) > 10-4
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From fractional to integral• Combined with the following result of Rödl, Schacht,
Siggers and Tokushige our result has consequences for integral packing.
• Let ν(H0,H) denote the maximum number of edge-disjoint copies of H0 in H (the H0-packing number of H).
• Let ν*(H0,H) denote the fractional relaxation.
• Trivially, ν*(H0,H) ≥ ν(H0,H).
• They proved that if H is an r-graph with n vertices then ν*(H0,H) < ν(H0,H) + o(nr).
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Corollaries for graphs
• If G is a graph with n vertices, andδ(G) > (1- 1/10k10)n then G has an asymptotically optimal Kk-packing.
• Same theorem holds for k-vertex graphs.• For triangles (k=3), δ(G) > 0.9999n suffices.
• The previously best known bound (for the missing degree) in the triangles case was 10-24 (Gustavsson).The previously best known bound for Kk was 10-37k-94 (Gustavsson).
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Corollary for 3-graphs
• If H is a 3-graph with n vertices andminimum co-degree (1-216-k)n then H has an asymptotically optimal K(k,3)-packing.
• Same theorem holds for k-vertex 3-graphs.
• The previously best known bound (for the missing co-degree) was 0 (The Rödl nibble).
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Tools used in the proof• Some linear algebra.
• Kahn’s Theorem:For every r* > 1 and every γ > 0 there exists a positive constant ρ=ρ(r*,γ) such that the following statement is true:
If U is an r*-graph with:(i) maxdeg < D(ii) maxcodeg < ρDthen there is a proper coloring of the edges of U with at most (1+γ)D colors.
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Tools used in the proof
• Several probabilistic arguments
large deviation,
local lemma.
• Hall’s Theorem for hypergraphs (by Aharoni and Haxell. Has a topological proof):
Let U={U1,…,Um} be a family of p-graphs. If for every W U there is a matching in UU WU of size greater than p(|W|-1) then U has an SDR.
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The proofRecall our goal:
There exists a positive constant α=α(k,r) so that every r-graph in which every (r-1)-set is contained in at least n(1-α) edges has a fractional K(k,r)-decomposition.
• Let t=k(r+1) and consider the family of the 3 r-graphs:F(k,r) = { K(k,r), K(t,r), H(t,r) }
where H(t,r) is a K(t,r) missing one edge.
• Lemma: K(k,r) fractionally decomposes each element of F(k,r). (To show that K(k,r) fractionally decomposes H(t,r) requires some work. We use linear algebra here.)
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The proof
For r=2 (graphs) it suffices to take t=2k-1 and the lemma is easy.
Example: r=2, k=3 hence t=5 and F(3,2) = { K(3,2), K(5,2), H(5,2) }
1 2
a cb
H(5,2)
w(i,x,y)=1/2
i=1,2
x,y=a,b,c
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The proof
It suffices to prove the stronger theorem
There exists a positive constant α=α(k,r) so that everyr-graph in which every (r-1)-set is contained in at leastn(1-α) edges has an integral F(k,r)-decomposition.
• Let ε = ε(k,r) be chosen later.
• Let η = (2-H(ε)0.9)1/ε. H(ε) the entropy function.
• Let α = min{ (η/2)2 , ε2/(t24t+1) }
• Let γ satisfy (1-αt2t)(1-γ)/(1+γ)2 > 1-2αt2t
• Let r* =
• Let ρ = ρ(r*,γ) be the constant from Kahn’s theorem.
• Assume n is suff. large as a function of all these constants.
r
t
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The proof
• Let δd(H) and Δd(H) denote the min and maxd-degrees of H, 0 < d < r.
• Our r-graph H satisfies δd(H) >
• It is not difficult to prove (induction) that every edge of H lies on many K(t,r). In fact, if c(e) denotes the number of K(t,r) containing e then
)1(
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nec
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n
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The proof
• Color the edges of H randomly usingq=n1/(4r*-4) colors (that’s many colors!)
• Let Hi be the spanning r-graph colored with i.
• Easy (Chernoff): δd(Hi) very close to δd(H)/q
• Not so easy: we would also like to show thatci(e) is very close to its expectation c(e)n-1/4.
Note that two K(t,r) that contain e may share other edges as well – a lot of dependence.
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The proof
• By considering the dependency graph of the c(e) events we can show:
(1+γ)nt-r-1/4 > ci(e) (t-r)! > (1-γ)nt-r-1/4(1-αt2t)
• We fix the coloring with q colors satisfying the above.
• For each Hi we create another r*-graph Ui:- the vertices of Ui are the edges of Hi
- the edges of Ui are the copies of K(t,r) in Hi
• Notice that Δ(Ui) < D = (1+γ)nt-r-1/4 (t-r)!-1
Notice that Δ2(Ui) < nt-r-1 << ρD
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The proof• By Kahn’s theorem this means that the K(t,r) copies of Hi
can be partitioned into at most D(1+γ) packings.
• We pick one of these packings at random. Denote it Li.
• The set L=L1 U…U Lq is a K(t,r) packing of H.
• Let M denote the edges of H not belonging to any element of L.
• Let p = A p-subset {S1,…,Sp} of L is good fore M if we can select one edge from each Si such that, together with p, we have a K(k,r).
• We say that L is good if for each e M we can select a good p-subset, and all |M| selections are disjoint.
1
r
k
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Example: being good
ab
b
c
L M
a
c
S100
S700
{S100,S700} is good for (a,c)
k=3
r=2
So:
t=5
p=2
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The proof• Recall F(k,r) = { K(k,r), K(t,r), H(t,r) }.
Clearly:L is good → H has an Fk-decomposition.
• It remains to show that there exists a good L. We will show that with positive probability, the random selection of the q packings L1 U…U Lq yields a good L.
• We use Hall’s theorem for hypergraphs.
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The proof• Let M={e1,…,em}.
• Let U={U1,…,Um} be a family of p-graphs defined as follows:
• The vertex set of Ui is L (i.e, K(t,r) copies)
• The edges of Ui are the p-subsets of L good for ei
• U has an SDR → L is good.
• Thus, it suffices to show that the random selection of the q packings L1 U…U Lq guarantees that, with positive probability, for every W U there is a matching in UU WU of size greater than p(|W|-1).
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The proof
• It turns out that the only thing needed to guarantee this is to show that with positive probability, for all 1 < d < r:Δd(H[M]) < 2ε
• Once this is established, the remainder of the claim is deterministic, namely
• Δd(H[M]) < 2ε → U has an SDR.(purely combinatorial proof, but not so easy).
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dr
dn
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Open problems• Determine the correct value of α(k,r).
• The simplest case is α(3,2) (triangles). We currently have α(3,2) > 10-4.
• A construction shows that α(3,2) ≤ ¼.
• More generally, a construction given in the paper shows that α(k,2) ≤ 1/(k+1).We conjecture α(k,2) = 1/(k+1).
• For hypergraphs we don’t even know what to conjecture.