Transcript Geometric Graphs

Geometric Graphs
Moshe Rosenfeld
University of Washington,
Tacoma
&
Vietnam National University
Hanoi University of Science
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Geometric Graphs
By geometric graphs we mean
graphs G(M,D) where M is a metric
space and D is a subset of R+.
Every graph is a geometric graph,
every graph is a “faithful” subgraph
of G(R2 , N).
In this talk I will survey some
examples that “bothered” me and
as fit for such an occasion solicit
help to solve some open problems.
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Triangle free graphs.

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
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A quick history of this problem.

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
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To conclude this 25 years long journey
with Jarik Nesetril
We “documented” this exploration
that was guided by Paul Erdős
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During this journey, other
problems surfaced.
For example, what is the
largest number of lines through
the origin in Rd such that
among any three lines there is
an orthogonal pair?
Can the graphs constructed
yield estimates for the Ramsey
number R(3,3,…,3)?
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Probably the most famous geometric
graph is the unit distance graph G(R2,
1).
This problem asks for the
smallest number of colors
needed to color the points of
the plane R2 so that two
points at distance 1 apart
receive different colors.
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How does a problem become
famous?
What made this problem famous?
• It is 62 years old.
• Very simple to understand
• At least five mathematicians were “credited”
with it: Edward Nelson, Paul ErdÖs, Hugo
Hadwiger, Leo Moser and Martin Gardener
• The lower and upper bounds (4,7) were
established in 1950.
• No progress since!
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What made it famous?
This is Hadwiger’s 7-coloring of G(R2 ,1)
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Inside is Moser’s spindle showing that at
least four colors are needed,
a 3-color transfer.
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 The attractions of this problem are
many.
 It is very easy to understand.
 It is finite in nature, all we need to find
is a finite 7 chromatic graph that can
be embedded in G(R2 , 1).
 Subgraphs of G(R2,1) with large girth.
 Many variations, such as choosability,
circular chromatic number, color sets
missing one color, other set axioms…
 G(Rd , 1), G(R2 , D),
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It can get “ugly.” This is a graph of order 45, girth 5,
chromatic number 4 embedded in G(R2 ,1)
This is Hadwiger’s 7-coloring of G(R2 ,1)
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Inside is Moser’s spindle showing that at
least four colors are needed,
a 3-color transfer.
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It follows that if D  R is bounded and
min(D)   > 0 then (G(R2 , D) is
finite.
What if D is not bounded?
This brings us to the odd-distance
graph: G(R2 , {1,3,5,7,…})
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Odd distances in R2
• The six distances determined by 4 points in R2
cannot be all odd.
• Putnam 1992
• R. Graham, B. Rothschild, E. Strauss proved:
the maximum number of points in Rd such that
all distances among them is an odd integer is
d+1 unless d = 14 + 16k.
In these dimensions we can have d + 2 points.
An elementary classroom proof
X2 Y2
X1 Y1
X3 Y3
o
 r r1,2 r1,3 
 x1 y1 


  x1 x2 x3  
2


  r2,1 r2 r2,3 
 x2 y2   

y
y
y
1
2
3 

2 
x y 
r
r
r
3
,
1
3
,
2
3 
 3 3

2
1
2
1
r
2 r2,1
r1, 2
r22
r1,3
r2,3  0
2
3
r3,1 r3, 2
But :
r
2ri 2  2(2ki  1) 2  2 mod 8
2
And: 2ri, j  ri  r  vi vj 1mod8
2
2
j
2 1 1
1 2 1  4  0 mo d8
1 1 2
The odd-distance graph
• In 1994 in Boca Raton I asked Paul ErdŐs and
Herbert Wilf whether R2 can be colored in a
finite number of colors so that two points at
odd integral distance have distinct colors?
(obvious lower bound 4)
• ErdŐs also asked what is the maximum
number of odd integral distances among n
points in R2
Density, typical ErdŐs questions.
• Given n points in R2, how many distances can
be 1? (ErdŐs, 1946).
• How many distances can be odd?
• How many times can the largest distance occur
among n points in R2?
• A “biological proof.”
• Clearly, the maximum number of odd
distances among n points in R2 is  t4(n)
(Turán’s number)
• L. Piepemeyer proved that Kn,n,n can be
embedded in R2 so that all edges have odd
integral distance.
Isosceles triangle side 7
Rotate once
Distances: 3, 5, 8
Realizes K2,2,2
P1
The embedding of
K3,3,3 :
3 equilateral triangles
with side 72.
P1, Y2, R3 sides 16,
56, 56
P3, Y1, R2 same
Y1
P2, Y3,R1 same
All other edges are:
49 (equilateral
triangles), 21, 35 and
39
R1
Y2
R2
P3
P2
R3
Y3
What makes it work.
 32 + 52 + 3.5 = 72
 212 + 352 + 21.35 = 492
 But also 392 + 162 = 492
 In general, we proved that a2 + b2 + a.b = 72n
has a solution with gcd(a,7) = 1. Which
implies that it has n different solutions {a,b}.
 This allows us to rotate an equilateral triangle
of side length 72n n+1 times about its center
and obtain an embedding of Kn+1,n+1,n+1 in
G(R2,Odd).
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Collateral benefits
 As an aside, we also obtain a set of 3n points in
R2, no three on a line, such that all distances
among them are integers.
 Haiko Harborth was first to use this construction
in search of the smallest diameter of n points in
R2, no three on a line with all distances integers.
 We believe that our approach will help us find
sets with smaller diameter.
 We are also looking at sets of points in general
position (no 3 on a line no 4 on a circle) with
pairwise integral distances.
 Current record is 9.
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Can the odd-distance graph compete with
the unit-distance graph for fame?
• Hayri Ardal (SFU)
• Jano Manuch (SFU)
• Moshe Rosenfeld (UWT)
• Saharon Shelah (Hebrew University)
• Ladislav Stacho (SFU)
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Some notable subgraphs of the
odd-distance graph.
• The integral lattice is 2-colorable
• The rational points are 2-colorable.
• You cannot go from one rational point to
every other point making odd integral
jumps stepping only on rational points.
• Every 3-colorable graph is a subgraph of
G(R2,odd).
• G(R2,odd) is not k-list colorable for any
integer k.
Theorem: The chromatic
number of G(R2 , odd) is  5.
• Construct a 4-color transfer.
• Key: “120o Pythagorean triples”:
a2 + b2 + ab = c2
(3, 5, 7)
THE
ARMS SPINDLE
21 points in R2 that
require 5 colors.
Choosability
• The unit distance graphs in R2 and R3
are countably choosable.
• We proved: The R2 odd-distance
graph is countably choosable.
• The R3 odd-distance graph is not
countably choosable.
G(R3 ,odd) is not countably
choosable
Bn = (0,0,4n2 + 4n)
{(x,y,0) | x2 + y2 =1}
L((x,y,0)) = A  N,
|A| = 0
L(Bn) = {n, n+1, …}
The circular chromatic
number of G(R2 ,odd)  4.5
(Xuding et. al.)
Can we prove that
G(R2 ,odd)  5 ?
Most likely it is!
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A terminal, logical inexactitude.
Peter Brass, William Moser and
Janos Pach wrote in their book
Research Problems in Discrete
Geometry :
..the existence of a K4 is the only
obstruction. That is, every finite K4 free graph can be represented by
odd distances in the plane. (p. 252)
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Odd wheels
W-5 is not a
Subgraph of
G(R2,odd)
Nam Le Tien
and M. R.
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The main tool is this matrix or the
similar nx2 matrix.
X2 Y2
X1 Y1
X3 Y3
o
 r r1,2 r1,3 
 x1 y1 


  x1 x2 x3  
2


  r2,1 r2 r2,3 
 x2 y2   

y
y
y
1
2
3 

2 
x y 
r
r
r
3
,
1
3
,
2
3 
 3 3

2
1
Is the chromatic number
of G(R2,odd) finite?
• Interestingly, G(R2,odd) has no finite
Lebesgue measurable coloring.
• This follows immediately from a
theorem of Furstenberg, Katznelson and
Weiss [FKW] which asserts that for
every measurable subset A  R2 with
positive upper density, there exists a
number r0 so that A contains a pair of
points at distance r for every r > r0 .
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Steps ina the proof
1
a2
a5
a4
a3
1. <oai, oai+2> are all rational or all are irrational.
2. If all are rational then we can multiply everything
by an odd integer so all 2<oai, oaj> are integers.
3. Every integer 2<oai, oai+2> is restricted in its
value mod 4 (only two values are possible).
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4. We may assume that
<oa1,oa3> = <oa1,oa4> mod 4
One of the two possibilities is <oa1,oa3> = 3
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HELP!!!
1. Odd wheels are not subgraphs of
G(R2,odd).
2. Are there triangle–free graphs that
are not subgraphs of G(R2,odd)
3. Can every 3-coloarble graph be
faithfully embedded in G(R2,odd)
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Problems
Find a 6-chromatic subgraph of G(R2,odd)
Find a subgraph of G(R2,odd) with circular
chromatic number  5.
Find lower bounds for G(R3,odd)
Is Kn,n,n,n a subgraph of G(R3,odd)
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Problems
Does G(R3,odd) have 5 chromatic subgraphs
with large girth?
Is there an infinite, unbounded set D for
which (G(R2, D) < ?
There is an uncountable dense subgraph of
G(R2, odd) with chromatic number 2. Is there
such a set whose vertices form a “fat” nonmeasurable set with finite chromatic number?
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Thank you
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