Existence of a spanning cyclic ladder graph in a balanced

Existence of a spanning cyclic ladder graph
in a balanced bipartite graph
Peter Heinig
Introduction
The Bachelor's Thesis(continued)
It is common in mathematics that rushing in does not succeed. If, to sketch an example,
This leads to the (as yet unproved) conjecture that there is the following
f : D
!C
is a holomorphic function on an open disk
D
the boundary of
is
not
D
C
, and every point on
a singular point (that means that around every point of
there is an open disk and a holomorphic continuation of
holomorphically extended to an open disk
~
D
f
to that disk), then
f
strengthening in the case of bipartite balanced graphs. Note that the
@D
minimum degree in the hypothesis has been halved.
> 0 and every 2 N there exists =
(; ) > 0 and n0 = n0(; ) such that every balanced bipartite
1
2
1
1
graph G with jG j n0 and (G ) 2 2 + jG j = 4 + jG j
contains as a subgraph every balanced bipartite graph H with jH j = jG j
and (H ) and bw(H ) jH j.
G
My result proves a special case of this conjecture because H := CL 2
not only has the proportionally small bandwidth of at most jH j but
r
even constant bandwidth, since 8r 2 N4 : bw CL
= 4.
can be
Conjecture. For every
having the same midpoint and a strictly
D, but it is not possible to prove this in general by greedily applying
the local continuation property to run around the circle @D , since in general there is
no control over the radii of the disks around the points of @D . However, it is easy to
larger radius than
prove it by arguing in an appropriate and more global way (an example can be found
in [2], Theorem 2.19.).
j
Analogously, there is a deep global principle in modern combinatorics which in its
formulation for graphs essentially states that for literally
every
graph, if only it is
large enough (the numbers of vertices that are known to be sucient turn out to
be extremely large but they can be explicitly stated, albeit normally not in decimal
j
The second way is to view the result as a generalization of the fact
notation), certain global preparatory provisions can be made, which allow one to prove
statements (mostly about the existence of certain subgraphs) which otherwise seem
(which is known at least since 1963, see [1]) that for a balanced bipartite
impossible to prove by greedily and locally appealing to the given hypotheses, these
graph
hypotheses often being of a very local kind, like minimum degree conditions. The most
G
jG j + 1 implies the existence of
a minimum degree of at least
4
+1 term is replaced
a Hamilton cycle. In the case of my theorem, the important keyword here is Szemerédi's regularity lemma , about which more can be
by a learned in sources like [9],[8],[3],[4],[10],[11],[12].
+ jG j
term,
>0
being an arbitrarily small constant, i.e. the
excess minimum degree is allowed to be arbitrarily small
To reiterate a point from above, one of the most interesting aspects of the regularity
but gets arbitrarily large
lemma is that there are several facts whose formulation looks quite clean and far
Gj
CL 2
relative
to
jG j
absolutely . In return for this stronger hypothesis
j
removed from the rather technical statements concomitant with the regularity lemma,
the subgraph
but no one has a clue how to prove them without it. One example is the proof of
a Hamilton cycle. For example, by counting the Hamilton cycles that
a conjecture of Seymour that a graph with
least
k n
k +1
must contain as a subgraph the
n
Gj
CL 2 alone contains, regardless of the many extra edges thehost graph
G has got, it follows that a minimum degree of at least 14 + jG j implies
(when jG j is large enough) not only the existence of one Hamilton cycle
jG j Hamilton cycles if jG j 2 ( mod 4 ) and of jG j + 2
but of
2
2
Hamilton cycles if jG j 0
( mod 4 ), i.e. a non-constant number of
j
vertices and a minimum degree of at
k th power of a Hamilton cycle (see [6] for
more).
The Bachelor's Thesis
distinct Hamilton cycles.
The main result of my thesis, which was supervised by Anusch Taraz
Note that these minimum degree conditions are very near to being
and Julia Böttcher, is a proof for the following statement.
> 0 there exists n0 = n0( ) 2 N
such that every balanced bipartite graph G with jG j n0 and (G ) 1 + jG j contains CL G2 as a subgraph.
4
Here, jG j denotes the number of vertices of G , being `balanced' means
best possible since a minimum degree of
G
Theorem [Heinig 2008]. For every
j
CL
Gj
2 is dened to be K 2
Gj
C2
j
two vertices,
jG j
4
does not even imply that
is connected (why?), let alone that it contains a Hamilton cycle.
j
A glance at the proof
that the two classes of the bipartite graph have equal cardinality, and
j
being forced to appear is much more structured than
a cycle
Gj
CL 2
j
Gj
The gure below is a schematic overview of how
C 2 where K 2 denotes a complete graph with
jG j vertices, and the cartesian product
with
2
j
into
G.
First
G
gets embedded
is globally prepared using the regularity lemma (and
Gj
CL 2
j
much additional tweaking), then
gets embedded at one fell swoop
This picture shows a
using the so-called Blow-Up Lemma (see [5]). The diculty mostly
CL 2 in
of jG j =
lies in (1) the fact that the regularity lemma method invariably involves
Gj
j
drawing
the
64,
der
of
case
ignoring
rungs.
It
ample
for
with
32
an
ex-
is
relatively
tiny
but
absolutely
really prove the existence of a subgraph which has
large
number
of
exactly
as many
vertices as the graph it exists in, and (2) the minimum degree condition
type
which
of graph whose exis-
would
tence as a spanning
tttttttttttttttttttttttttttttttttttttttttttt
x x
x
x x
x
x x
x x
subgraph is proved in
tttttttttttttttttttttttttttttttttttttttttttt
my thesis.
tttttttttttttttttttttttttttttttttttttttttttt
the
arbitrarily
waste vertices which have to be re-distributed skillfully in order to
i.e. a cyclic ladgraph
a
11 12
is
barely
not
1s1
above
be
a
threshold
sucient.
21 22
2s2
31 32
which
(as
y1
1
y1s
1
y1
s1 −1
y2
1
y2s
2
y2
s2 −1
mentioned)
tttttttttttttttttttttttttttttttttttttttttttt
x3s
3
41 42
tttttttttttttttttttttttttttttttttttttttttttt
of graphs.
already
y3
1
y3s
3
y3
s3 −1
y4
1
tttttttttttttttttttttttttttttttttttttttttttt
x4s
4
x` x`
1 2
y4s
4
y4
s4 −1
x`
y`
1
s`
x1
1
x1
2
y`
s`
y`
s` −1
There are two good ways to put this result in context and the rst
of them will be outlined now. In [13] the authors prove the following
theorem, where
(G )
bandwidth of the graph
denotes the chromatic number and bw
Selected References
(G ) the
G (standard notions whose denitions are easily
[1] John Moon, Leo Moser: On hamiltonian bipartite graphs. Israel Journal of Mathematics 1: 163-165 (1963)
[2] Aleksei Ivanovich Markushevich: Theory of Functions of a Complex Variable, Volume 1, Revised English Edition,
Translated and Edited by Richard A. Silverman, Prentice-Hall, Inc., 11th printing (1966)
[3] Miklós Simonovits, Vera T. Sós: Szemerédi's Partition and Quasirandomness. Random Structures and Algorithms,
Vol. 2, No. 1: 1-10 (1991)
[4] Noga Alon, Richard Alter Duke, Hanno Lefmann, Vojtech Rödl, Raphael Yuster: The Algorithmic Aspects of the
Regularity Lemma. Journal of Algorithms 16: 80-109 (1994)
[5] János Komlós, Gábor Sárközy, Endre Szemerédi: Blow-Up Lemma. Combinatorica 17: 109-123 (1997)
[6] János Komlós, Gábor N. Sárközy, Endre Szemerédi: Proof of the Seymour conjecture for Large Graphs. Annals
of Combinatorics, Volume 2, pp. 42-60 (1998)
[7] Sarmad Abbasi: How Tight is the Bollobás-Komlós Conjecture? Graphs and Combinatorics 16: 129-137 (2000)
[8] Reinhard Diestel: Graph Theory. Springer-Verlag, Graduate Texts in Mathematics, Volume 173, Third Edition
(2005)
[9] Jozef Skokan: Regularity Lemma Basics. (not published in printed form; easily found on the web)(2006)
[10] Terence Tao: Szemerédi's Regularity Lemma revisited. Contributions to Discrete Mathematics 1, 8-28 (2006)
[11] Ben Joseph Green, Terence Tao: Szemerédi's Theorem. Scholarpedia, 2(7):3446 (2007)
[12] Terence Tao: What is good mathematics? Bulletin (New Series) of The American Mathematical Society, Volume
4, Number 4 : p. 628-634 (2007)
[13] Julia Böttcher, Mathias Schacht, Anusch Taraz: Proof of the bandwidth conjecture of Bollobás and Komlós.
Mathematische Annalen, Volume 343, Number 1, Pages 175-205 (2009)
found).
> 0 and for every r 2 N and for every 2 N
there exists = (; r; ) > 0 and n0 = n0 (; r; ) such that every
r
1
graph G with jG j n0 and (G ) r + jG j contains as a subgraph
every graph H with jH j = jG j and (H ) r and (H ) and
bw(H ) jH j.
Theorem. For every
[13] Julia Böttcher, Mathias Schacht, Anusch Taraz:
Chair of Applied Geometry and Discrete Mathematics (M9)
j
Boltzmannstr. 3
j
85748 Garching
j
www-m9.ma.tum.de