Transcript Asymptotically optimal Kk-packings of dense graphs via fractional Kk-decompositions Raphael Yuster University of Haifa Fractional decompositions of dense hypergraphs Raphael Yuster University of Haifa.

Asymptotically optimal
Kk-packings of dense
graphs via fractional
Kk-decompositions
Raphael Yuster
University of Haifa
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Fractional
decompositions of
dense hypergraphs
Raphael Yuster
University of Haifa
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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.
• K(k,r) will denote the complete r-graph with k
vertices. We think of k and r as fixed.
• We prove (for n > n0):
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.
• In fact:
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 Haxell,
Nagle and Rödl, 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).
• If H is an r-graph with n vertices (r=2,3) it has
been proved by Haxell, Nagle and Rödl that
ν*(H0,H) < ν(H0,H) + o(nr).
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Corollaries for graphs
• If G is a graph with n vertices, and
δ(G) > (1- 0.1k -10)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).
• However, Gustavsson guarantees a
decomposition in case the appropriate
divisibility conditions holds.
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Corollary for 3-graphs
• If H is a 3-graph with n vertices and
minimum 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 (Rödl).
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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:
maxdeg < D
maxcodeg < ρD
then there is a proper coloring of the edges of U with at
most (1+γ)D colors.
• Several probabilistic arguments.
• Hall’s Theorem for hypergraphs by Aharoni
and Haxell (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 proof
Recall the 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) Consider the 3 r-graphs:
F(k,r) = { K(k,r), K(t,r), H(t,r) }
H(t,r) is a K(t,r) missing one edge.
• 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. Here
we use some linear algebra.)
• For r=2 it suffices to take t=2k-1 and the proof
is easy. E.g. K5- fractionally decomposes K3.
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The proof – cont.
It suffices to prove the stronger theorem
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 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*
t 
=  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.
• Let δd(H) and Δd(H) denote the min and max
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d-degrees of H, 0 < d < r, resp.
The proof – cont.
n  d 
(1   )
• Our r-graph H satisfies δd(H) > 
r d 
• 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
nt-r > c(e) (t-r)! > nt-r(1-αt2t)
• Color the edges of H randomly using
q=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 that
ci(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 – cont.
• Still, we can prove it by partitioning the c(e)
events to many (but not too many) subsets such
that all events in the same part are independent,
show large deviation on each part and the sum
of slacks is still negligible. Thus,
(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 as
follows:
- 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 – cont.
• 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 by 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 =
•
k 
   1
A p-subset
 r {S1,…,Sp}
of L is good for e 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.
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Example: being good
k=3
b
a
r=2
S100
So:
t=5
a
p=2
L
b
M
c
S700
c
{S100,S700} is good for (a,c)
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The proof – cont.
• 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.
• 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 edge set of Ui are the p-subsets of L that
are good for ei
• U has an SDR → L is good.
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The proof – cont.
• 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).
• 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  n  d 
r d 
• Once this is established, the remainder of
the claim is deterministic, namely
n  d 
2ε r  d  →


• Δd(H[M]) <
U has an SDR.
(purely combinatorial proof, but not so
easy).
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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.
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