Asymptotically optimal Kk-packings of dense graphs via fractional Kk-decompositions

Abstract

Let H be a fixed graph. A fractional H-decomposition of a graph G is an assignment of nonnegative real weights to the copies of H in G such that for each e∈E(G), the sum of the weights of copies of H containing e is precisely one. An H-packing of a graph G is a set of edge disjoint copies of H in G. The following results are proved. For every fixed k>2, every graph with n vertices and minimum degree at least n(1-1/9k10)+o(n) has a fractional Kk-decomposition and has a Kk-packing which covers all but o(n2) edges.

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arXiv:math/0311449v1 [math.CO] 25 Nov 2003
Asymptotically optimal Kk-packings of dense graphs via fractional
Kk-decompositions
Raphael Yuster∗
Department of Mathematics
University of Haifa at Oranim
Tivon 36006, Israel
Abstract
Let H be a fixed graph. A fractional H-decomposition of a graph G is an assignment of
nonnegative real weights to the copies of H in G such that for each e ∈ E(G), the sum of
the weights of copies of H containing e in precisely one. An H-packing of a graph G is a set
of edge disjoint copies of H in G. The following results are proved. For every fixed k > 2,
every graph with n vertices and minimum degree at least n(1 − 1/9k10) + o(n) has a fractional
Kk-decomposition and has a Kk-packing which covers all but o(n2) edges.
1Introduction
All graphs considered here are finite, undirected and simple. For standard graph-theoretic termi-
nology the reader is referred to [1]. Let H be a fixed graph. For a graph G, the H-packing number,
denoted νH(G), is the maximum number of pairwise edge-disjoint copies of H in G. A function
ψ from the set of copies of H in G to [0,1] is a fractional H-packing of G if?
number, denoted ν∗
H(G), is defined to be the maximum value of |ψ| over all fractional packings
ψ. Notice that, trivially, ν∗
H(G) ≥ νH(G). In case νH(G) = e(G)/e(H) we say that G has an
H-decomposition. In case ν∗
H(G) = e(G)/e(H) we say that G has a fractional H-decomposition. It
is well known that computing νH(G) is NP-Hard for every fixed graph H with more than two edges
in some connected component [3]. It is well known that computing ν∗
time for every fixed graph H as this amounts to solving a (polynomial size) linear program.
The combinatorial aspects of the H-packing and H-decomposition problems have been studied
extensively. Wilson in [11] has proved that whenever n ≥ n0= n0(H), and Knsatisfies two obvious
e∈Hψ(H) ≤ 1 for
each e ∈ E(G). For a fractional H-packing ψ, let |ψ| =?
H∈(G
H)ψ(H). The fractional H-packing
H(G) is solvable in polynomial
∗e-mail: raphy@research.haifa.ac.il World Wide Web: http:\\research.haifa.ac.il\˜raphy
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necessary divisibility requirements, Knhas an H-decomposition. The H-packing problem for Kn
(n ≥ n(H)) was solved [2], by giving a closed formula for ν(H,Kn). For graphs G which are
not complete, giving sufficient conditions guaranteeing an H-decomposition or, at least, a packing
covering all but a small fraction of the edges seems to be an extremely difficult task, even if we
assume that G is dense. To be more precise, let us define the following problem and parameter.
Problem: Given a fixed graph H, determine cH, which is the supremum of all possible con-
stants c guaranteeing that every graph G with n vertices and minimum degree δ(G) ≥ (1−c)(n−1)
has νH(G) ≥ (1 − on(1))e(G)/e(H).
Wilson’s Theorem implies that cH ≥ 0 exists.
R¨ odl’s result [10] which is weaker than Wilson’s for graphs, but is more general since it applies
to r-uniform hypergraphs as well. Gustavsson [5] has proved that cH > 0 for every graph H.
However, Gustavsson’s lower bound for cHis horribly close to 0. Already for H = K3it only gives
cK3> 10−24, and, more generally, if H has k vertices then cH > 10−37k−94. Gustavsson’s result
does however, show that the minimum degree requirement, together with necessary divisibility
conditions, guarantees an H-decomposition. The exact value of cHis unknown for any fixed non-
bipartite graph H. Notice that, trivially, cH= 1 in case H is a bipartite graph as the Tur´ an number
of such graphs is o(n2). However, if we insist on having a packing of size ⌊e(G)/e(H)⌋ then it is
known that a minimum degree of 0.5n(1 + on(1)) suffices for each fixed bipartite graph H (other
than the trivial K2) having a vertex of degree one (this includes all trees) and this is asymptotically
tight [12].
In this paper we prove the first reasonable general lower bound for cH.
In fact, cH ≥ 0 can also be derived from
Theorem 1.1 For all k ≥ 3, cKk≥ 1/9k10.
Although Theorem 1.1 is stated only for Kk, a simple argument given in the sequel shows that the
same lower bound holds for any graph H with k vertices. Theorem 1.1 is deduced as a combination
of two powerful theorems, the first of which is the following.
Theorem 1.2 For all k ≥ 3, any graph G with n vertices and δ(G) ≥ n(1 − 1/9k10) + o(n) has a
fractional Kk-decomposition.
The proof of Theorem 1.2 appears in the following section.
Recently, Haxell and R¨ odl [6] proved that the H-packing number and the fractional H-packing
number are very close for dense graphs. A simpler and more general proof of their result appears
in [13].
Theorem 1.3 [Haxell and R¨ odl [6]] For any fixed graph H, if G has n vertices then ν∗
νH(G) = o(n2).
H(G) −
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Theorem 1.2 gives that for sufficiently large n, any graph G with n vertices and δ(G) ≥ n(1 −
1/9k10)+o(n) has ν∗
Kk(G) = e(G)/e(Kk). Thus, by Theorem 1.3, it also has νKk(G) ≥ e(G)/e(Kk)−
o(n2) = (1 − on(1))e(G)/e(Kk). Consequently, cKk≥ 1/9k10and Theorem 1.1 follows.
Finally, we note that an 1/(k + 1) upper bound for cKkis given in the final section together
with some additional concluding remarks.
2Proof of Theorem 1.2
Let F be a fixed family of graphs. An F-decomposition of a graph G is a set L of subgraphs
of G, each isomorphic to an element of F, and such that each edge of G appears in precisely
one element of L. Let K−
t
denote the complete graph with t vertices, missing one edge. Let
Fk= {Kk, K2k−1, K−
theorem.
2k−1}. The proof of Theorem 1.2 is a corollary of the following stronger
Theorem 2.1 For all k ≥ 3, every graph with n vertices and minimum degree at least n(1 −
1/9k10) + o(n) has an Fk-decomposition.
The following simple lemma shows that Theorem 1.2 is a corollary of Theorem 2.1.
Lemma 2.2 For all k ≥ 2, the graphs K2k−1and K−
2k−1have a fractional Kk-decomposition.
Proof: It is trivial that for all k′≥ k, Kk′ has a fractional Kk-decomposition. In Particular, K2k−1
has a fractional Kk-decomposition. Let A = {u,v} denote the set of the two non-adjacent vertices
of K−
2k−1, and let B denote the set of the remaining 2k − 3 vertices. Each edge incident with A
lies on?2k−4
contain a vertex of A. Since
k−2
k−3
Kkcontaining a vertex of A, and assigning the value 0 to the remaining copies of Kk, we obtain a
fractional Kk-decomposition of K−
2k−1.
k−2
?copies of Kk. Each edge with both endpoints in B lies on 2?2k−5
k−3
?copies of Kkthat
k−2
to each copy of
?2k−4
?
= 2?2k−5
?, by assigning the value 1/?2k−4
?
We now focus on proving Theorem 2.1. For the rest of this section, let t = 2k − 1. Notice that
the o(n) term in the statement of Theorem 2.1 allows us to assume, whenever necessary, that n is
sufficiently large. In the proof of Theorem 2.1 it will be convenient to use Wilson’s Theorem [11]
mentioned in the introduction. (We note that it is also possible to use R¨ odl’s packing theorem [10]
instead of Wilson’s Theorem at the price of some complication in the proof). Wilson’s Theorem
applied to Ktstates the following.
Lemma 2.3 [Wilson] Let t > 2 be a positive integer. There exists N = N(t) such that for all
n > N with n ≡ 1,t mod t(t − 1), there is a decomposition of Kninto Kt.
We shall prove Theorem 2.1 under the relaxed assumption that n ≡ 1 mod t(t−1). We first need
to justify this relaxation. Indeed, if 1 < b ≤ t(t − 1) and n ≡ b mod t(t − 1) then we can perform
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the following preprocessing. Let v be any vertex of G, and let N(v) denote its neighborhood.
Let G[N(v)] be the subgraph induced by this neighborhood. Notice that G[N(v)] has less than n
vertices, but has minimum degree at least n(1 − 2/9k10) + o(n). By the theorem of Hajnal and
Szemer´ edi [7], such a high minimum degree for a graph with at most n vertices is far more than what
is needed in order to guarantee that G[N(v)] has a spanning subgraph G′with each component of
G′being either a Kk−1or a K3k−4. In fact, a minimum degree of at least n(1−1/(3k −4)) already
guarantees the existence of such a G′. Now, if H is a Kk−1component of G′then H∪{v} is a Kkcopy
of G. If H is a K3k−4component of G′then H ∪v contains two edge-disjoint subgraphs, one being
Kkand the other being K−
t, and with all 3k − 4 edges between v and the vertices of H absorbed.
We thus have a set of edge-disjoint subgraphs of G, each being either Kkor K−
all edges incident with v. Deleting v and the edges of these subgraphs we remain with a graph with
n−1 vertices and minimum degree at least δ(G)−(3k−4) ≥ (n−1)(1−1/9k10)+o(n−1). Repeating
this process at most b − 1 times we eventually have a graph with n′= n − b + 1 ≡ 1 mod t(t − 1)
vertices and minimum degree at least δ(G)−(3k−4)((2k−1)(2k−2)−1) ≥ n′(1−1/9k10)+o(n′),
which satisfied our relaxed assumption. Our preprocessing shows that any Fk-decomposition of this
resulting n′-vertex graph can be extended to an Fk-decomposition of the original n-vertex graph.
t, and that absorb
Proof of Theorem 2.1 We may assume that n ≡ 1 mod t(t − 1) and that, whenever necessary,
n is sufficiently large as a function of k (and hence t). In particular, n > N(t) where N(t) is the
constant from Lemma 2.3. Fix a Kt-decomposition of Kn. Namely, let D be a family of t-sets
of [n] = {1,...,n} such that each pair appears in precisely one element of D. Such a D is also
called a t-design. Notice that |D| =
?n
Sπ = {π(j) : j ∈ S}. Hence, Dπ = {Sπ : S ∈ D} is also a t-design. Let G be an n-vertex
graph with vertex set [n]. For a permutation π of [n] let Gπbe the family of?n
subgraphs of G whose elements are the induced subgraphs of G on Sπ, for all S ∈ D. Notice that if
G = Knthen, trivially, Gπis a Kt-decomposition for each π, but if G ?= Kn, Gπcontains elements
that are not isomorphic to Kt. The following is a simple corollary of Lemma 2.3.
2
?/?t
2
?. For a permutation π of [n], and for S ∈ D, let
2
?/?t
2
?edge-disjoint
Corollary 2.4 Let 0 < α < 1 be fixed. Let G be a graph with n > N(t) vertices, n ≡ 1 mod t(t−1).
If δ(G) ≥ (1 − α)n and π is any permutation of [n] then Gπhas at least (1 − o(1))n2(
elements isomorphic to Ktand at most (1 + o(1))n2(α
that are not isomorphic to Kt.
1
t(t−1)−α
2)
2)(?t
?n
2
?− 1) edges appear in elements of Gπ
?− (1 − α)n2/2. Thus, Gπ has at most
2
?− (1 − α)n2/2)(?t
Proof: The number of non-edges of G is at most
?n
elements of Gπ.
2
2
?−(1−α)n2/2 elements that are not Ktand therefore Gπhas at least?n
?/?t
2
?−?n
2
?+(1−α)n2/2
elements isomorphic to Kt and at most (?n
2
2
?− 1) edges of G are in non-Kt
Assume that δ(G) ≥ n(1−1/9k10)+ o(n). Our goal is to show that there exists a permutation
π, such that Gπhas some “nice” properties. Let Aπdenote the set of edges of G that appear in
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non-Ktelements of Gπ. By Corollary 2.4, with α = 1/9k10,
|Aπ| ≤ (1 + o(1))n2
?
1
18k10
???t
2
?
− 1
?
≤ (1 + o(1))n2
9k8. (1)
Consider the spanning subgraph of G consisting of the edges of Aπ. It will not be confusing to
denote this subgraph by Aπas well. Let Fπ⊂ Gπbe the set of Kt-elements of Gπ. Put r =?k
We say that an r-subset S = {H1,...,Hr} of Fπis good for e ∈ Aπif we can select edges fi∈ Hi
such that {f1,...,fr,e} is the set of edges of a Kkin G. We say that π is good if for each e ∈ Aπ
there exists an r-subset S(e) of Fπ such that S(e) is good for e and such that if e ?= e′then
S(e) ∩ S(e′) = ∅.
2
?−1.
Lemma 2.5 If π is good then G has an Fk-decomposition.
Proof: For each e ∈ Aπ, pick a copy of Kkin G containing e and precisely one edge from each
element of S(e). As each element of S(e) is a Kt, deleting one edge from such an element results
in a K−
disjoint. The remaining element of Fπnot belonging to any of the S(e) are each a Kt, and they
are edge-disjoint from each other and from the previously selected Kkand K−
t. We therefore have |Aπ| copies of Kk and |Aπ|(?k
2
?− 1) copies of K−
t, all being edge
t.
Our goal in the remainder of this section is, therefore, to show that there exists a good π.
We use probabilistic and counting arguments to derive this fact. We will show that with positive
probability, a randomly selected π is good. We begin by showing that with high probability, a
randomly selected π has the property that Aπhas a relatively small maximum degree.
Let v be any vertex of G, and let E(v) denote the set of edges incident with v.
assumption, |E(v)| ≥ (1−1/9k10)n. Let Eπ(v) = E(v)∩Aπ. Notice that if π is selected uniformly
at random then |Eπ(v)|, which is the degree of v in Aπ, is a random variable. We say that a subset
S ⊂ E(v) is separated by π if each edge of S belongs to a different element of Gπ. Let β = 3/k8
and consider any fixed set S ⊂ E(v) with |S| = ⌊βn⌋. We shall prove that the probability that
S ⊂ Eπ(v) and that S is separated by π is much smaller than the total number of subsets of E(v)
with size ⌊βn⌋. Thus, the probability that the degree of v in Aπexceeds tβn is also very small (in
fact, much smaller than 1/n), and consequently, the maximum degree of Aπis at most tβn almost
surely. Let S = {e1,...,em} where m = ⌊βn⌋ and let ei= (v,vi). Let Sπ(i) denote the element
of Gπto which the edge eibelongs, i = 1,...,m. Notice that S is separated by π if and only if
Sπ(i) ?= Sπ(j) for all i ?= j. Clearly, by using conditional probabilities we have
By our
Pr[(S ⊂ Eπ(v)) ∧ (S is separated by π)] = (2)
m
?
i=1
Pr[(ei∈ Aπ)∧(∀j < i, Sπ(i) ?= Sπ(j)) | ({e1,...,ei−1} ⊂ Aπ)∧(Sπ(j) ?= Sπ(j′), 1 ≤ j < j′< i)].
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