The odd-distance graph

To be or not to be…famous

• Hayri Ardal (SFU)

• Jano Manuch (SFU)

• Moshe Rosenfeld (UWT)

• Saharon Shelah (Hebrew University)

• Ladislav Stacho (SFU)

The unit distance graph

The geometry Junkyardhttp://www.ics.uci.edu/~eppstein/junkyard/geom-color.html

Geometric Graph Coloring ProblemsThese problems have been extracted from "Graph Coloring Problems", T. Jensen and B. Toft, Wiley 1995. See that book

(specifically chapter 9, on geometric and combinatorial graphs) or its online archives for more information about them.

•Hadwiger-Nelson Problem. Let G be the infinite graph with all points of the plane as vertices and having xy as an edge if and only if the points x and y have distance 1. What is the chromatic number of G? It is known to be at least four and at most seven.

                                                                                                  

The Moser Spindle

What makes this problem famous?

• It is 60 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!

What made it famous?

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 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

o

X1 Y1

X2 Y2

X3 Y3

232,31,3

3,22

21,2

3,12,12

1

321

321

33

22

11

rrr

rrr

rrr

yyy

xxx

yx

yx

yx

022

32,31,3

3,22

21,2

3,12,12

1

rrr

rrr

rrr

8mod2)12(22: 22 ii krBut

8mod12:222

, jijiji vvrrrAnd

8mod04

211

121

112

The odd-distance graph

• In 1994 in Boca Raton I asked Paul Erdös 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)

• Erdös also asked what is the maximum number of odd integral distances among n points in R2

Density

• Given n points in R2, how many distances can be 1? (ErdÖs, 1946).

• How many times can the largest distance occur among n points in R2?

• A “biological proof.”

• Clearly, the maximum number of distances 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 7Rotate onceDistances: 3, 5, 8Realizes K2,2,2

The embedding of K(3,3,3): 3 equilateral triangles with side 72.P1, Y2, R3 sides 16, 56, 56 P3, Y1, R2 same P2, Y3,R1 sameAll other edges are: 49 (equilateral triangles), 21, 35 and 39

P1

P2

P3

Y1

Y2

Y3

R1

R2

R3

The magic matrix

• One matrix does it all!

Rotate an equilateral triangle with side 7n-1 n times to obtain an embedding of Kn,n,n or

A set of n3 points that maximizes the number of odd distances.

Surprisingly, on a single circle.

Some notable subgraphs of the odd-distance graph.

• The integral lattice is 2-colorable

• The rational points are 2-colorable.

• Every 3-colorable graph can be realized as an odd distance graph in R2

• The R2 odd distance graph is not k-list colorable for any integer k.

Theorem: The R2 odd distance graph requires at least 5 colors.

• 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.

Pseudo-Pythagorian Triples

There are many other odd distances hidden in the triangular lattice.

Any 120o triangle with two sides along the lattice will yield an odd distance.

Here is a small sample:

3, 5, 7 (32 + 52+ 3*5 = 72)

7, 33, 37 (72 + 332 + 7*3 = 372)

11, 85, 91 (112+ 852 + 11*85 = 912)

13, 35, 43 (132 + 452 + 13*45 = 432)

17, 63, 73 (172 + 632 + 17*63 = 732)

Problem

Is every odd prime a member of a Pseudo-Pythagorian Triple?

The triangular lattice is 4-colorable

Choosability

• The unit distance graphs in R2 and R3 are countably choosable.

• The R2 odd-distance graph is countably choosable.

• The R3 odd-distance graph is not countably choosable.

R3 is not countably choosable

{(x,y,0) | x2 + y2 =1}

Bn = (0,0,4n2 + 4n)

L(Bn) = {n, n+1, …}L((x,y,0)) = A N,

|A| = 0

The odd distance graph in R2 is countably choosable.

Theorem: the integer-distance graph in R2 has the 0 property.

• For every finite set X = {x1,…,xk} define:

NxyxyXxxy

kifXXF

jiji ||||||,||,,:

1)(

Corollary: GN(R2), Godd(R2), G{1}(R2) are countably choosable.

Unit distance vs. Odd distance

Lower bound 4 5

Upper bound 7 0

Density Open Tn

(Turán number)

Forbidden subgraphs

Many, (G) > 7,K2,3

K4

Is the chromatic number of the odd-distance graph finite?

• Interestingly, the Odd Distance Graph has no finite measurable coloring.

• This follows immediately from a theorem of Furstenberg, Katznelson and Weiss [FKW] which asserts that for every Lesbesgue 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 .

Problems

• Is the odd distance chromatic number of a circle in R2 = 3?

• The R3 odd distance graph does not contain a K5 . Is Kn,n,n,n a subgraph of the R3 odd distance graph?

• Find lower bounds for the chromatic number of the R3 odd distance graph.

Problems

• Does Godd(R2) contain 5-chromatic subgraphs with arbitrary large girth?

• Let D = {0 < d1 < d2 < …} be an unbounded sequence of numbers. Is it true that for every mapping

f : R2 S1 and every > 0, there are two points p, q in R2 such that ||p – q|| D and ||f(p) – f(q)|| < ?

Summary

The Odd-Distance graph is a very simple object.

Offers challenging problems.

Applications? Surprisingly, there seem to be applications for the unit-distance graph in Quantum Mechanics.