Download From Rainbow to the Lonely Runner

From Rainbow to the
Lonely Runner
Daphne Liu
Department of Mathematics
California State Univ., Los Angeles
January 24, 2007
Overview:
Plane coloring
Distance
Graphs
Fractional
Chromatic
Number
Circular
Chromatic
Number
Lonely Runner
Conjecture
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: Vertices and edges.

A proper vertex coloring: A function that
assigns to each vertex a color so that
adjacent vertices receive different colors.
Chromatic number of G: The minimum
number of colors used in a proper vertex
coloring of G.
 (G)

Example
 (C3 )  3
 (C5 )  3
At least we need four colors
for coloring the plane
Assume three colors, red, blue and green, are
used.
X
Known Facts
 (C2n )  2;  (C2n1 )  3
4   ( , {1})  7;  ( , {1})  ?
2
2
[Moser & Moser, 1968; Hadweiger et al., 1964]
 (Q , {1})  2
2
[van Luijk, Beukers, Israel, 2001]
http://www.math.leidenuniv.nl/~naw/serie5/deel01/sep2000/pdf/problemen3.pdf
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.
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.
The circular chromatic number of G is the smallest
r such that there exists an r-coloring for G.
 c (G)
Example, C5
0
0
1
1.5
0.5
0.5
2
2
 c (C5 )  2.5
1.5
1
Known Results:
The following hold for any graph G:
 f (G)   c (G)   (G)
 c (G)    (G)
1
 c (C 2n 1 )  2 
n
→2
 c (C 2n )   (C 2n )  2
Distance Graphs
Eggleton, 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) 
| D | 1
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
L. & Zhu, J. Graph Theory, 2004
Zhu, 2001
1 ?
 f (G(Z, D))   c (G(Z, D)) 
 | D | 1
 (D)
||
Lonely Runner Conjecture
1
Chang, L., Zhu, 1999
 (D)
Density of Sequences w/
Missing Differences

Let D be a set of positive integers.
Example, D = {1, 4, 5}. => μ ({1, 4, 5}) = 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, …}
“density” of this M(D) is 1/3.

μ (D) = maximum density of an M(D).
Theorem & Conjecture
(L & Zhu, 2004, JGT)

If D = {a, b, a+b} and gcd(a, b)=1, then
 2a  b   2b  a 
 3   3 
 (D)  Max {
,
}
2a  b
2b  a
[Conj. by Rabinowitz & Proulx, 1985]
Example: μ ({1, 4, 5}) = Max{ 1/3, 1/3 } = 1/3
Example: μ ({3, 5, 8}) = Max{ 2/11, 4/13 } = 4/13
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
(k  1)m
 ( D) 
4(k  1)m  1
Example: μ ({2, 3, 5, 8}) = ?