Embedding spanning subgraphs of small bandwidth.

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Embedding spanning subgraphs of small
bandwidth
Julia B¨ ottcher1
Zentrum Mathematik, Technische Universit¨ at M¨ unchen, Boltzmannstraße 3,
D-85747 Garching bei M¨ unchen, Germany
Mathias Schacht2
Institut f¨ ur Informatik, Humboldt-Universit¨ at zu Berlin, Unter den Linden 6,
D-10099 Berlin, Germany
Anusch Taraz1
Zentrum Mathematik, Technische Universit¨ at M¨ unchen, Boltzmannstraße 3,
D-85747 Garching bei M¨ unchen, Germany
Abstract
In this paper we prove the following conjecture by Bollob´ as and Koml´ os: For every
γ > 0 and positive integers r and ∆, there exists β > 0 with the following property. If
G is a sufficiently large graph with n vertices and minimum degree at least (r−1
and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum
degree at most ∆, then G contains a copy of H.
r+γ)n
Keywords: extremal graph theory, spanning subgraphs, regularity lemma
1Email:{boettche|taraz}@ma.tum.de
2Email:schacht@informatik.hu-berlin.de
The first and third author were supported by DFG grant TA 309/2-1. The second author
was supported by DFG grant SCHA 1263/1-1.

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1Introduction and results
One of the fundamental results in extremal graph theory is the theorem by
Erd˝ os and Stone [6] which implies that any fixed graph H of chromatic number
r is forced to appear as a subgraph in any sufficiently large graph G if the
average degree of G is at least (r−2
constant γ.
In this extended abstract we prove a similar result for spanning subgraphs
H of small bandwidth that was conjectured by Bollob´ as and Koml´ os. It is
obvious that for a spanning graph H, it no longer suffices to guarantee that
G has a large average degree, since we need (to be able to control) every
single vertex of G, and thus we shift our emphasis to a large minimum degree
instead. Also, it is clear that in this regime the lower bound has to be raised
at least to δ(G) ≥r−1
r-partite graph with partition classes almost, but not exactly, of the same size
(thus G has minimum degree almostr−1
of vertex disjoint r-cliques.
There are a number of results where a minimum degree ofr−1
sufficient to guarantee the existence of a certain spanning subgraph H. A well
known example is Dirac’s theorem [5]. It asserts that any graph G on n vertices
with minimum degree δ(G) ≥ n/2 contains a Hamiltonian cycle. Another
classical result of that type by Corr´ adi and Hajnal [4] states that every graph
G with n vertices and δ(G) ≥ 2n/3 contains ⌊n/3⌋ vertex disjoint triangles.
This was generalised by Hajnal and Szemer´ edi [7], who proved that every
graph G with δ(G) ≥
cliques, each of size r.
P´ osa and Seymour [14] suggested a further extension of this theorem. They
conjectured that, at the same threshold δ(G) ≥r−1
fact contain a copy of the (r − 1)-st power of a Hamiltonian cycle (where the
(r−1)-st power of an arbitrary graph is obtained by inserting an edge between
every two vertices of distance at most r − 1 in the original graph). This was
proven in 1998 by Koml´ os, S´ ark¨ ozy, and Szemer´ edi [12] for sufficiently large n.
Recently, several other results of a similar flavour have been obtained which
deal with a variety of spanning subgraphs H, such as, e.g., trees, F-factors,
and planar graphs (see the survey [13] and the references therein). In an
attempt to move away from results that concern only graphs H with a special,
rigid structure, Bollob´ as and Koml´ os [9, Conjecture 16] conjectured that every
r-chromatic graph on n vertices of bounded degree and bandwidth at most
o(n), can be embedded into any graph G on n vertices with δ(G) ≥ (r−1
r−1+ γ)n, for an arbitrarily small positive
rn: simply consider the example where G is the complete
rn) and let H be the spanning union
rn is indeed
r−1
rn must contain a family of ⌊n/r⌋ vertex disjoint
rn, such a graph G must in
r+γ)n.

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(A graph is said to have bandwidth at most b, if there exists a labelling of
the vertices by numbers 1,...,n, such that for every edge {i,j} of the graph
we have |i − j| ≤ b.) In this extended abstract we present a proof of this
conjecture.
Theorem 1.1 For all r,∆ ∈ N and γ > 0, there exist constants β > 0 and
n0∈ N such that for every n ≥ n0the following holds.
If H is an r-chromatic graph on n vertices with ∆(H) ≤ ∆ and bandwidth
at most βn and if G is a graph on n vertices with minimum degree δ(G) ≥
(r−1
r+ γ)n, then G contains a copy of H.
The analogue of Theorem 1.1 for bipartite graphs H was announced by
Abbasi [1] in 1998, and a proof based on our methods can be found in [8].
In [3] we proved the 3-chromatic case of this theorem. One central ingredient
to the proof was the existence of the square of a Hamiltonian cycle in graphs of
high minimum degree as asserted by the affirmative solution of the conjecture
of P´ osa mentioned above. However, it turned out that the (r −1)-st power of
a Hamiltonian cycle is not well connected enough to carry over these methods
to the r-chromatic case.
The following simple example shows that the statement of Theorem 1.1
becomes false when the bandwidth condition on H is dropped. Let H be a
random bipartite graph with bounded maximum degree and partition classes
of size n/2 each, and let G be the graph formed by two cliques of size (1/2+γ)n
each, which share exactly 2γn vertices. It is then easy to see that G cannot
contain a copy of H, since in H every set of vertices of size (1/2 − γ)n has
more than 2γn external neighbours.
Also, the γ term in the minimum degree condition on G is necessary in the
following sense: Abbasi [2] showed that if γ → 0 and ∆ → ∞ then β must
tend to 0 in Theorem 1.1. However, the bound on β coming from our proof is
rather poor, having a tower-type dependence on 1/γ.
Let us finally address the rˆ ole of the chromatic number of H in Theo-
rem 1.1. In the same way that the Hamiltonian cycle on an odd number of
vertices is forced as a spanning subgraph in any graph of minimum degree
n/2 (although it is 3- and not 2-chromatic), other (r + 1)-chromatic graphs
are forced already when δ(G) ≥ (r−1
in [10], it seems that the crucial question here is whether all r +1 colours are
needed by many vertices.
The following extension of Theorem 1.1 tries to go into a somewhat similar
direction. Assume that the vertices of H are labelled 1,...,n. For two positive
integers x,y, a proper (r + 1)-colouring σ : V (H) → {0,...,r} of H is said
r+ γ)n. As already observed by Koml´ os

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to be (x,y)-zero free with respect to such a labelling, if for each t ∈ [n] there
exists a t′with t ≤ t′≤ t + x such that σ(u) ?= 0 for all u ∈ [t′,t′+ y].
Theorem 1.2 For all r,∆ ∈ N and γ > 0, there exist constants β > 0 and
n0∈ N such that for every n ≥ n0the following holds.
Let H be a graph with ∆(H) ≤ ∆ whose vertices are labelled 1,...,n such
that, with respect to this labelling, H has bandwidth at most βn, an (r + 1)-
colouring that is (8rβn,4rβn)-zero free, and uses colour 0 for at most βn
vertices in total.
If G is a graph on n vertices with minimum degree δ(G) ≥ (r−1
then G contains a copy of H.
r
+ γ)n,
We conclude with the remark that our proof is constructive and yields a
polynomial time algorithm, which finds an embedding of H in G if H is given
along with a valid r-colouring (respectively, (r +1)-colouring) and a labelling
of the vertices respecting the bandwidth bound βn.
2 Outline of the proof
Roughly speaking, the proof of Theorem 1.2 is split into two main lemmas.
While they deal exclusively with the graph G and the graph H respectively,
they are linked to each other in the following way: the lemma for G suggests
a partition of G and communicates the structure of this partition (but not the
graph G) to the lemma for H . The lemma for H then tries to find a partition
of H with a very similar structure, and returns the sizes of the partition classes
to the lemma for G. The latter then adjusts its partition classes by shifting a
few vertices of G, until they fit exactly the class sizes of H.
The initial partition constructed by the lemma for G is obtained using the
regularity lemma of Szemer´ edi [15]. This lemma guarantees that the vertex
set of every graph G can be partitioned in such a way that most of its edges
belong to sufficiently “random–like” induced bipartite graphs (so–called ε-
regular pairs).
Once compatible partitions of G and H have been found via the lemma
for G and the lemma for H, respectively, we find an embedding of H in G
with the help of the blow-up lemma of Koml´ os, S´ ark¨ ozy, and Szemer´ edi [11].
This lemma asserts that r-chromatic spanning graphs of bounded degree can
be embedded into the union of r classes that form
minimum degree dn for some small constant d.
?r
2
?
ε-regular pairs with

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