2007 | OriginalPaper | Chapter
General Balanced Subdivision of Two Sets of Points in the Plane
Authors : M. Kano, Miyuki Uno
Published in: Discrete Geometry, Combinatorics and Graph Theory
Publisher: Springer Berlin Heidelberg
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Let
R
and
B
be two disjoint sets of red points and blue points, respectively, in the plane such that no three points of
R
∪
B
are co-linear. Suppose
ag
≤ |
R
| ≤ (
a
+ 1)
g
,
bg
≤ |
B
| ≤ (
b
+ 1)
g
. Then without loss of generality, we can express |
R
| =
a
(
g
1
+
g
2
) + (
a
+ 1)
g
3
, |
B
| =
bg
1
+ (
b
+ 1)(
g
2
+
g
3
), where
g
=
g
1
+
g
2
+
g
3
,
g
1
≥ 0,
g
2
≥ 0,
g
3
≥ 0 and
g
1
+
g
2
+
g
3
≥ 1. We show that the plane can be subdivided into
g
disjoint convex polygons
$X_{1}\cup \cdots \cup X_{g_1}\cup Y_{1}\cup \cdots \cup Y_{g_2}\cup Z_{1}\cup \cdots \cup Z_{g_3}$
such that every
X
i
contains
a
red points and
b
blue points, every
Y
i
contains
a
red points and
b
+ 1 blue points and every
Z
i
contains
a
+ 1 red points and
b
+ 1 blue points.