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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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