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 Junkyard Geometric Graph Coloring Problems These problems have been extracted from "Graph Coloring Problems", T. Jensen and B. Toft, Wiley See that book (specifically chapter 9, on geometric and combinatorial graphs) or its online archives for more information about them.online archives 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 Erd Ö s, Hugo Hadwiger, Leo Moser and Martin Gardener The lower and upper bounds (4, 7) were established in No progress since!

What made it famous?

Odd distances in R 2 The six distances determined by 4 points in R 2 cannot be all odd. Putnam 1992 R. Graham, B. Rothschild, E. Strauss the maximum number of points in R d such that all distances among them is an odd integer is d+1 unless d = k. In these dimensions we can have d + 2 points.

An elementary classroom proof o X 1 Y 1 X 2 Y 2 X 3 Y 3

The odd-distance graph In 1994 in Boca Raton I asked Paul Erdös and Herbert Wilf whether R 2 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 R 2

Density Given n points in R 2, how many distances can be 1? ( Erd Ö s, 1946). How many times can the largest distance occur among n points in R 2 ? A “biological proof.”

Clearly, the maximum number of distances is  t 4 (n) (Turán’s number) L. Piepemeyer proved that K n,n,n can be embedded in R 2 so that all edges have odd integral distance.

Isosceles triangle side 7 Rotate once Distances: 3, 5, 8 Realizes K 2,2,2

The embedding of K(3,3,3): 3 equilateral triangles with side 7 2. P1, Y2, R3 sides 16, 56, 56 P3, Y1, R2 same P2, Y3,R1 same All other edges are: 49 (equilateral triangles), 21, 35 and 39 P1P1 P2P2 P3P3 Y1Y1 Y2Y2 Y3Y3 R1R1 R2R2 R3R3

The magic matrix One matrix does it all! Rotate an equilateral triangle with side 7 n-1 n times to obtain an embedding of K n,n,n or A set of n 3 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 R 2 The R 2 odd distance graph is not k-list colorable for any integer k.

Theorem: The R 2 odd distance graph requires at least 5 colors. Construct a 4-color transfer. Key: “120 o Pythagorean triples”: a 2 + b 2 + ab = c 2 (3, 5, 7)

THE ARMS SPINDLE 21 points in R 2 that require 5 colors.

Pseudo-Pythagorian Triples There are many other odd distances hidden in the triangular lattice. Any 120 o triangle with two sides along the lattice will yield an odd distance.

Here is a small sample: 3, 5, 7 ( *5 = 7 2 ) 7, 33, 37 ( *3 = 37 2 ) 11, 85, 91 ( *85 = 91 2 ) 13, 35, 43 ( *45 = 43 2 ) 17, 63, 73 ( *63 = 73 2 )

Problem Is every odd prime a member of a Pseudo- Pythagorian Triple? The triangular lattice is 4-colorable

Choosability The unit distance graphs in R 2 and R 3 are countably choosable. The R 2 odd-distance graph is countably choosable. The R 3 odd-distance graph is not countably choosable.

R 3 is not countably choosable {(x,y,0) | x 2 + y 2 =1} B n = (0,0,  4n 2 + 4n) L(B n ) = {n, n+1, …} L((x,y,0)) = A  N, |A| =  0

The odd distance graph in R 2 is countably choosable.

Theorem: the integer-distance graph in R 2 has the  0 property. For every finite set X = { x 1,…,x k } define: Corollary: G N (R 2 ), G odd (R 2 ), G {1} (R 2 ) are countably choosable.

Unit distance vs. Odd distance Lower bound45 Upper bound7 00 DensityOpenT n (Turán number) Forbidden subgraphs Many,  (G) > 7, K 2,3 K4K4

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  R 2 with positive upper density, there exists a number r 0 so that A contains a pair of points at distance r for every r > r 0.

Problems Is the odd distance chromatic number of a circle in R 2 = 3? The R 3 odd distance graph does not contain a K 5. Is K n,n,n,n a subgraph of the R 3 odd distance graph? Find lower bounds for the chromatic number of the R 3 odd distance graph.

Problems Does G odd (R 2 ) contain 5-chromatic subgraphs with arbitrary large girth? Let D = {0 < d 1 < d 2 < …} be an unbounded sequence of numbers. Is it true that for every mapping f : R 2  S 1 and every  > 0, there are two points p, q in R 2 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.