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Zdenek Dvorak
Comparable box dimension
Commits
dfb16030
Commit
dfb16030
authored
Sep 08, 2021
by
Zdenek Dvorak
Browse files
The subgraph argument.
parent
0951065d
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dfb16030
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@@ 306,23 +306,92 @@ Moreover, $x\prec y$, implying that $E_i(u)\cap E_i(v)\neq\emptyset$ for $i=0,\l
$
xy
\in
E
(
T
_
\beta
)
$
. Hence, again we have
$
f
(
u
)
\cap
f
(
v
)
\neq\emptyset
$
.
\end{proof}
Note that in Theorem~
\ref
{
thmcs
}
, we only get a representation of a supergraph
of
$
G
$
.
%We will need the fact that the chromatic number is at most exponential in the comparable box dimension;
%this follows from~\cite{subconvex} and we include the argument to make the dependence clear.
%\begin{lemma}\label{lemmachrom}
%If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$colorable.
%\end{lemma}
%\begin{proof}
%We actually show that $G$ is $(3^d1)$degenerate. Since every induced subgraph of $G$ also
%has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$
%is less than $3^d$. Let $v$ be a vertex of $G$ such that $f(v)$ has the smallest volume. For every neighbor $u$ of $v$,
%there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$.
%Note that $f(v)\cup \bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in
%a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+N(v)\le 3^d$.
%\end{proof}
We can now combine Theorem~
\ref
{
thmcs
}
with Lemma~
\ref
{
lemmacliq
}
.
\begin{corollary}
\label
{
corcs
}
If
$
G
$
is obtained from graphs in a class
$
\GG
$
by cliquesums, then there exists a graph
$
G'
$
such that
$
G
\subseteq
G'
$
and
$
\cbdim
(
G'
)
\le
(
\cbdim
(
\GG
)+
1
)
\bigl
(
2
^{
\cbdim
(
\GG
)
}
+
1
\bigr
)
\le
6
^{
\cbdim
(
\GG
)
}$
.
\end{corollary}
Note that only bound the comparable box dimension of a supergraph
of
$
G
$
. To deal with this issue, we show that the comparable box dimension of a subgraph
is at most exponential in the comparable box dimension of the whole graph.
This is essentially Corollary~25 in~
\cite
{
subconvex
}
, but since the setting is somewhat
different and the construction of~
\cite
{
subconvex
}
uses rotated boxes,
we provide details of the argument.
A
\emph
{
star coloring
}
of a graph
$
G
$
is a proper coloring such that any two color classes induce
a star forest (i.e., a graph not containing any 4vertex path). The
\emph
{
star chromatic number
}
$
\chi
_
s
(
G
)
$
of
$
G
$
is the minimum number of colors in a star coloring of
$
G
$
.
We will need the fact that the star chromatic number is at most exponential in the comparable box dimension;
this follows from~
\cite
{
subconvex
}
and we include the argument to make the dependence clear.
\begin{lemma}
\label
{
lemmachrom
}
If
$
G
$
has a comparable box representation
$
f
$
in
$
\mathbb
{
R
}^
d
$
, then
$
G
$
has star chromatic number at most
$
2
\cdot
9
^
d
$
.
\end{lemma}
\begin{proof}
Let
$
v
_
1
$
,
\ldots
,
$
v
_
n
$
be the vertices of
$
G
$
ordered nonincreasingly by the size of the boxes that represent them;
i.e., so that
$
f
(
v
_
i
)
\sqsubseteq
f
(
v
_
j
)
$
whenever
$
i>j
$
. We greedily color the vertices in order, giving
$
v
_
i
$
the smallest
color different from the colors of all vertices
$
v
_
j
$
such that
$
j<i
$
and either
$
v
_
jv
_
i
\in
E
(
G
)
$
, or there exists
$
m>j
$
such that
$
v
_
jv
_
m,v
_
mv
_
i
\in
E
(
G
)
$
. Note this gives a star coloring: A path on four vertices always contains a 3vertex subpath
$
v
_{
i
_
1
}
v
_{
i
_
2
}
v
_{
i
_
3
}$
such that
$
i
_
1
<i
_
2
,i
_
3
$
, and in such a path, the coloring procedure gives each vertex a distinct color.
Hence, it remains to bound the number of colors we used. Let us fix some
$
i
$
, and let us first bound the number of vertices
$
v
_
j
$
such that
$
j<i
$
and there exists
$
m>i
$
such that
$
v
_
jv
_
m,v
_
mv
_
i
\in
E
(
G
)
$
. Let
$
B
$
be the box that is five times larger than
$
f
(
v
)
$
and has the same center as
$
f
(
v
)
$
. Since
$
f
(
v
_
m
)
\sqsubseteq
f
(
v
_
i
)
\sqsubseteq
f
(
v
_
j
)
$
, there exists a translation
$
B
_
j
$
of
$
f
(
v
_
i
)
$
contained in
$
f
(
v
_
j
)
\cap
B
$
. The boxes
$
B
_
j
$
for different
$
j
$
have disjoint interiors and their interiors are also disjoint from
$
f
(
v
_
i
)
\subset
B
$
, and thus the number of such indices
$
j
$
is at most
$
\vol
(
B
_
j
)/
\vol
(
f
(
v
_
i
))
1
=
5
^
d

1
$
.
A similar argument shows that the number of indices
$
m
$
such that
$
m<i
$
and
$
v
_
mv
_
i
\in
E
(
G
)
$
is at most
$
3
^
d

1
$
.
Consequently, the number of indices
$
j<i
$
for which there exists
$
m
$
such that
$
j<m<i
$
and
$
v
_
jv
_
m,v
_
mv
_
i
\in
E
(
G
)
$
is at most
$
(
3
^
d

1
)
^
2
$
.
Consequently, when choosing the color of
$
v
_
i
$
greedily, we only need to avoid colors of at most
$$
(
5
^
d

1
)
+
(
3
^
d

1
)
+
(
3
^
d

1
)
^
2
<
5
^
d
+
9
^
d<
2
\cdot
9
^
d
$$
vertices.
\end{proof}
Next, let us show a bound on the comparable box dimension of subgraphs.
\begin{lemma}
\label
{
lemmasubg
}
If
$
G
$
is a subgraph of a graph
$
G'
$
, then
$
\cbdim
(
G
)
\le
\cbdim
(
G'
)+
\chi
^
2
_
s
(
G'
)
$
.
\end{lemma}
\begin{proof}
As we can remove the boxes that represent the vertices, we can assume
$
V
(
G'
)=
V
(
G
)
$
.
Let
$
f
$
be a touching representation by comparable boxes in
$
\mathbb
{
R
}^
d
$
, where
$
d
=
\cbdim
(
G'
)
$
. Let
$
\varphi
$
be a star coloring of
$
G'
$
using colors
$
\{
1
,
\ldots
,c
\}
$
, where
$
c
=
\chi
_
s
(
G'
)
$
.
For any distinct colors
$
i,j
\in\{
1
,
\ldots
,c
\}
$
, let
$
A
_{
i,j
}
\subseteq
V
(
G
)
$
consist of vertices
$
u
$
of color
$
i
$
such that there exists a vertex
$
v
$
of color
$
j
$
such that
$
uv
\in
E
(
G'
)
$
and
$
uv
\not\in
E
(
G
)
$
.
Let us define a representation
$
h
$
by boxes in
$
\mathbb
{
R
}^{
d
+
\binom
{
c
}{
2
}}$
by starting from the representation
$
f
$
and,
for each pair
$
i<j
$
of colors, adding a dimension
$
d
_{
i,j
}$
and setting
$
h
(
v
)[
d
_{
i,j
}
]=[
1
/
3
,
4
/
3
]
$
for
$
v
\in
A
_{
i,j
}$
,
$
h
(
v
)[
d
_{
i,j
}
]=[
4
/
3
,

1
/
3
]
$
for
$
v
\in
A
_{
j,i
}$
,
and
$
h
(
v
)[
d
_{
i,j
}
](
v
)=[
1
/
2
,
1
/
2
]
$
otherwise. Note that the boxes in this extended representation are comparable,
as in the added dimensions, all the boxes have size
$
1
$
.
Suppose
$
uv
\in
E
(
G
)
$
, where
$
\varphi
(
u
)=
i
$
and
$
\varphi
(
v
)=
j
$
and say
$
i<j
$
. The boxes
$
f
(
u
)
$
and
$
f
(
v
)
$
touch.
We cannot have
$
u
\in
A
_{
i,j
}$
and
$
v
\in
A
_{
j,u
}$
, as then
$
G'
$
would contain a 4vertex path in colors
$
i
$
and
$
j
$
.
Hence, in any added dimension
$
d'
$
, at least one of
$
h
(
u
)
$
and
$
h
(
v
)
$
is represented by the interval
$
[
1
/
2
,
1
/
2
]
$
,
and thus
$
h
(
u
)[
d'
]
\cap
h
(
v
)[
d'
]
\neq\emptyset
$
. Therefore, the boxes
$
h
(
u
)
$
and
$
h
(
v
)
$
touch.
Suppose now that
$
uv
\not\in
E
(
G
)
$
. If
$
uv
\not\in
E
(
G'
)
$
, then
$
f
(
u
)
$
is disjoint from
$
f
(
v
)
$
, and thus
$
h
(
u
)
$
is disjoint from
$
h
(
v
)
$
. Hence, we can assume
$
uv
\in
E
(
G'
)
$
,
$
\varphi
(
u
)=
i
$
,
$
\varphi
(
v
)=
j
$
and
$
i<j
$
. Then
$
u
\in
A
_{
i,j
}$
,
$
v
\in
A
_{
j,i
}$
,
$
h
(
u
)[
d
_{
i,j
}
]=[
1
/
3
,
4
/
3
]
$
,
$
h
(
v
)[
d
_{
j,i
}
]=[
4
/
3
,

1
/
3
]
$
, and
$
h
(
u
)
\cap
h
(
v
)=
\emptyset
$
.
Consequently,
$
h
$
is a touching representation of
$
G
$
by comparable boxes in dimension
$
d
+
\binom
{
c
}{
2
}
\le
d
+
c
^
2
$
.
\end{proof}
Let us now combine Lemmas~
\ref
{
lemmachrom
}
and
\ref
{
lemmasubg
}
.
\begin{corollary}
\label
{
corsubg
}
If
$
G
$
is a subgraph of a graph
$
G'
$
, then
$
\cbdim
(
G
)
\le
\cbdim
(
G'
)+
4
\cdot
81
^{
\cbdim
(
G'
)
}
\le
5
\cdot
81
^{
\cbdim
(
G'
)
}$
.
\end{corollary}
Let us remark that an exponential increase in the dimension is unavoidable: We have
$
\cbdim
{
K
_{
2
^
d
}}
=
d
$
,
but the graph obtained from
$
K
_{
2
^
d
}$
by deleting a perfect matching has comparable box dimension
$
2
^{
d

1
}$
.
Corollaries~
\ref
{
corcs
}
and~
\ref
{
corsubg
}
now give the main result of this section.
\begin{corollary}
\label
{
corcomb
}
If
$
G
$
is obtained from graphs in a class
$
\GG
$
by cliquesums, then
$
\cbdim
(
G
)
\le
5
\cdot
81
^{
6
^{
\cbdim
(
\GG
)
}}$
.
\end{corollary}
\section
{
The product structure and minorclosed classes
}
...
...
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