From Rainbow to the Lonely Runner

Daphne LiuDepartment of Mathematics

California State Univ., Los Angeles

January 24, 2007

Overview:

Distance Graphs

Fractional Chromatic Number

Circular Chromatic Number

Lonely Runner Conjecture

Plane coloring

Plane Coloring Problem

Color all the points on the xy-plane so that any two points of unit distance apart get different colors.

What is the smallest number of colors needed to accomplish the above ?

Seven colors are enough [Moser & Moser, 1968]

< 1

Graphs and Chromatic Number

A graph G contains two parts:

A proper vertex coloring:

Chromatic number of G:

(G)

Vertices and edges.

A function that assigns to each vertex a color so that adjacent vertices receive different colors.

The minimum number of colors used in a proper vertex coloring of G.

Example

3 )(C 3 3 )(C 5

At least we need four colors for coloring the plane

Assume three colors, red, blue and green, are used.

X

Known Facts

3 )(C 2; )(C 12n2n

? {1}) ,( 7; {1}) ,( 4 22

2 ){1} ,(Q 2 http://www.math.leidenuniv.nl/~naw/serie5/deel01/sep2000/pdf/problemen3.pdf

[Moser & Moser, 1968; Hadweiger et al., 1964]

[van Luijk, Beukers, Israel, 2001]

Circular Chromatic Number

Let G be a graph. Let r be a real number and Sr be a circle on the xy-plane centered at (0,0) with circumference r.

The circular chromatic number of G is the smallest r such that there exists an r-coloring for G.

An r-coloring of G is a function f : V(G) => Sr such that for adjacent vertices u and v, the circular distance (shorter distance on Sr) between f(u) and f(v) is at least 1.

(G) c

Example, C5

0

1

20.5

1.5

0

0.5

11.5

2

2.5 )(C 5 c

Known Results:

(G) (G) (G) cf

(G) (G) c

n

1 2 )(C 12n c

The following hold for any graph G:

2 )(C )(C 2n2n c

→ 2

Distance GraphsEggleton, Erdős et. al. [1985 – 1987]

For a given set D of positive integers, the distance graph G(Z, D) has:

Vertices: All integers Z as vertices;

Edges: u and v are adjacent ↔ |u - v| є D

D = {1, 3, 4}

0 1 2 3 4 5 6 7 8

Lonely Runner Conjecture

Suppose k runners running on a circular field of circumference r. Suppose each runner keeps a constant speed and all runners have different speeds. A runner is called “lonely” at some moment if he or she has (circular) distance at least r/k apart from all other runners.

Conjecture: For each runner, there exists some time that he or she is lonely.

Parameter involved in the Lonely Runner Conjecture

For any real x, let || x || denote the shortest distance from x to an integer. For instance, ||3.2|| = 0.2 and ||4.9||=0.1.

Let D be a set of real numbers, let t be any real number:

||D t|| : = min { || d t ||: d є D}.

φ (D) : = sup { || D t ||: t є R}.

Example

D = {1, 3, 4} (Four runners)||(1/3) D|| = min {1/3, 0, 1/3} = 0

||(1/4) D|| = min {1/4, 1/4, 0} = 0

||(1/7) D|| = min {1/7, 3/7, 3/7} = 1/7

||(2/7) D|| = min {2/7, 1/7, 1/7} = 1/7||(3/7) D|| = min {3/7, 2/7, 2/7} = 2/7

φ (D) = 2/7 [Chen, J. Number Theory, 1991] ≥ ¼.

Wills Conjecture

For any D, 1 |D|

1 (D)

Wills, Diophantine approximation, in German, 1967. Betke and Wills, 1972. (Confirmed for |D|=3.) Cusick, View obstruction problem, 1973. Cusick and Pomerance, 1984. (Confirmed for |D| ≤ 4.) Bienia et al, View obstruction and the lonely runner,

JCT B, 1998. (New name.) Y.-G. Chen, On a conjecture in diophantine

approximations, I – IV, J. Number Theory, 1990 &1991.

(A more generalized conjecture.)

Relations

1 |D| (D)

1 D)) (G(Z, D)) (G(Z, c

f

?

(D)

1

| |

Lonely Runner Conjecture

Zhu, 2001

Chang, L., Zhu, 1999

L. & Zhu, J. Graph Theory, 2004

Density of Sequences w/ Missing Differences

Let D be a set of positive integers. Example, D = {1, 4, 5}.

“density” of this M(D) is 1/3.

A sequence with missing difference of D, denoted by M(D), is one such that the absolute difference of any two terms does not fall not in D.

For instance, M(D) = {3, 6, 9, 12, 15, …}

μ (D) = maximum density of an M(D).

=> μ ({1, 4, 5}) = 1/3.

Theorem & Conjecture (L & Zhu, 2004, JGT)

If D = {a, b, a+b} and gcd(a, b)=1, then

} a2b

3a2b

,b2a

3b2a

{Max (D)

[Conj. by Rabinowitz & Proulx, 1985]

Example: μ ({3, 5, 8}) = Max{ 2/11, 4/13 } = 4/13

Example: μ ({1, 4, 5}) = Max{ 1/3, 1/3 } = 1/3

M(D) = 0, 2, 4, 6, 13, 15, 17, 19, 26, . . . .

Conjecture [L. & Zhu, 2004]

Conjecture: If D = {x, y, y-x, y+x} where x=2k+1 and y=2m+1, m > k, gcd(x,y)=1, then

11)m4(k

1)m(k )(

D

Example: μ ({2, 3, 5, 8}) = ?