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Chromatic Number of
Distance Graphs Generated by
the Sets { 2, 3, x, y }
Daphne Liu and Aileen Sutedja
Department of Mathematics
California State Univ., Los Angeles
Distance Graphs
Eggleton, Erdős, Skilton [1985 – 1987]
Fix D a set of positive integers, the
distance graph G(Z, D) has:
Vertices: All integers Z
Edges: | u - v |  D
D = { 1, 3, 4 }
0
1
2
3
4
5
6
7
8
Example
D = {1, 3, 5, 7, 9}. Then
0
1
2
3
χ (D) = 2
4
5
6
7
8
3 – Element Sets
 Let D = { a, b, c }, a < b < c.
2 if a, b, c are odd;
4 if D = {1, 2, 3m}, or

 (D)  
c = a + b but b - a  1,2 (mod 3);

3 otherwise.
[Eggleton, Erdős, Skilton 1985]
[Chen, Huang, Chang 1997]
[Voigt, 1999]
[Zhu, 2002]
Chromatic Number of Distance Graphs
If D contains only odd numbers then
χ (D) ≤ 2.
If D contains no multiples of k, then
χ (D) ≤ k.
If D is a subset of prime numbers then
χ (D) ≤ 4.
4 – Element Prime Sets
For a prime set D = {2, 3, p, q},
χ (D) = 3 or 4.
Question: For a prime set D = {2, 3, p, q}.
Which sets D have χ (D) = 4 ?
Complete solutions on
4-element prime sets
Let D be a prime set, D={2, 3, p, q}. Then
χ (D) = 4 if and only if p, q are twin primes, or
(p, q) is one of the following:
(11,19), (11,23), (11, 37), (11, 41),
(17, 29), (23, 31), (23, 41), (29, 37).
[Eggleton, Erdős, Skilton 1990]
[Voigt and Walther, 1994]
General 4 – element Sets
D = { 2, 3, x, x+s }, x > 3.
Kemnitz and Kolberg determined the
chromatic number for all s < 10.
Voigt and Walther proved that
χ(D) = 3 if s ≥ 10 and x ≥ s2 – 6s +3.
Our aim: Completely solve this problem.
Theorems
If D = {1, 2, 3, 4m} or
D = {x, y, y-x, y+x}, x and y are odd.
Then χ (D) = 4.
[Kemnitz & Marangio]
[L. & Zhu]
Let |D| = 4. Then χ (D) ≤ 4 unless
D is the above two types.
[Barajas & Serra 2008]
Main Tools – Useful Results
Chang, L., Zhu, 1999
Zhu, 2001
 1 
1
  ( D)  

 ( D)

(
D
)


Let D = {2, 3, x, y}. Then
1
 ( D)    ( D)  3
3
1
 ( D)    ( D)  4
3
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 differences of D,
denoted by M(D), is one such that the
absolute difference of any two terms does
not fall 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 [Chang, L., Zhu, 1999]
For any finite set of integers D,
1
 f (G( D)) 
  (G( D)).
 (D)
χ f (G) is the fractional chromatic number.
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}.
κ (D) : = sup { || D t ||: t є R}.
Example
D = { 1, 3, 4 }
||(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
κ (D) = 2/7
Useful Lemmas – Lower Bounds
1
 ( D)    ( D)  4
3
Theorem: Let 0 < t <1. If for every
D-sequence S there exists some
n ≥ 0 such that S[n]/(n+1) ≤ t
then μ(D) ≤ t.
[Haralambis 1977]
Corollary: If μ(D) ≥ 1/3, then there exists a
D-sequence S so that
S[n]/(n+1) ≥ 1/3 for all n ≥ 0.
Upper Bounds
1
 ( D)    ( D)  3
3
For any D, κ (D) = m/n, where n is the sum
of two elements in D.
Idea
0
1/3
1/3
Theorem [L. and Sutedja 2011]
D = { 2, 3, x, x+10 }.
4 if x  5;
 (D)  
3 otherwise.
D = { 2, 3, 6, y }.  (D)  4 if y  0,  1, 4 (mod 9);

3 otherwise.
D = { 2, 3, 10, y }
4 if y  0,  1 (mod 6);
 (D)  
D = { 2, 3, 4, y }.
3 otherwise.
Alternative Definition of κ (D)
1
Let   ( 0, ). For a positive integer m, let
2
I m ( )  { t  (0, 1) : || tm ||   }. Then
1
 (D)  sup {   ( 0, ) :  I m ( )  }.
2 mD
Let D  {2, 3, x, y}.
If I 2 (1 / 3)  I3 (1 / 3)  I x (1 / 3)  I y (1 / 3)   ,
1
Then  (D)  .
3
Theorem [L. and Sutedja 2011]
Let D ={ 2, 3, x, y }. Then χ(D)=3 for:
 X = 12, 13, 14, or x ≥ 17, and y ≥ 2 x
 X ≥ 53, y ≥ x - 11
 X = 9, y ≥ 36
 X = 11, y ≥ 66
 X = 15, y ≥ 60
 X = 16, y ≥ 48
Theorem (continue)
Let D = { 2, 3, x, x+s}, x ≥ 7 and s ≥ 11.
Then χ (D) = 3, except (x, x+s) is one of:
(9, 23), (11, 23), (11, 27), (11, 28), (11, 32),
(11, 37), (11, 41), (11, 46), (15, 41), (16, 37),
(17, 29), (18, 31), (23, 36), (23, 41), (24, 37),
(28, 41),
for which χ (D) = 4.
Open Problems
 Let D be a prime set with |D| = 5.
For what sets D, we have χ (D) = 3?
 Let D be a set with |D| = 4.
For what sets D, we have χ (D) = 3?
 Exact values of κ (D) and μ (D) for
D = { 2, 3, x, y }.
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 (Wills, Goddyn): For each
runner, there exists some time that he or
she is lonely.
Wills Conjecture
1
For any D,  (D) 
| D | 1
 Wills, Diophantine approximation, in German, 1967.
 Cusick and Pomerance, 1984. (True for |D| ≤ 4.)
 Chen, J. Number Theory, a generalized conjecture.
 Bohman, Holzman and Kleitman 2001. |D| = 5.
 Barajas & Serra 2008. |D| = 6.