Coloring Parameters of Distance Graphs

Daphne Liu

Department of Mathematics

California State Univ., Los Angeles

Overview:

Distance Graphs

Fractional Chromatic Number Lonely

Runner Conjecture

Plane coloring

Circular Chromatic Number

Plane Coloring Problem

What is the smallest number of colors to color all the points on the xy-plane so that any two points of unit distance apart get different colors?

G(R2, {1}) = Unit Distance Graph of R2.

χ (G(R2, {1})) = χ (R2, {1}) = ?

4 ≤ χ (R2, {1}) ≤ 7

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

< 1

At least we need four colors for coloring the planeAssume only use three colors: red, blue and green.

X

1

Rational Points on the Plane

2 ){1} ,(Q 2

http://www.math.leidenuniv.nl/~naw/serie5/deel01/sep2000/pdf/problemen3.pdf

[van Luijk, Beukers, Israel, 2001]

Distance Graphs (Eggleton, Erdős, Skelton 1985 - 1987)

Defined on the real line: Given a set D of

positive reals called forbidden set: G(R, D) has R as the vertex set u ~ v ↔ |u – v| D.

(Integral) Distance Graphs: Given a set D of

positive reals called forbidden set: G(Z, D) has Z as the vertex set u ~ v ↔ |u – v| D.

D = {1, 3, 4}

0 1 2 3 4 5 6 7 8

Example

Note: For any D, χ (G(Z, D)) ≤ |D| + 1.

Chromatic number of G(Z, P)

D = P, set of all primes. Then χ (G(Z, P)) = 4. [Eggleton et. al. 1985]

This problem is solved for |D| = 3, 4.

[Eggleton et al 1985]

[Voigt and Walther 1994]

Open Problem: For what D P,

χ (G(Z, D)) = 4 ?

Fractional Chromatic Number χf (G):

Give a weight, real in [0,1], to each independent set in G so that for each vertex v the total weights (of the independent sets containing v) is at least 1.

The minimum total weight of all the independent sets is the fractional chromatic number of G.

Facts on Fractional Chromatic Number

number. ceindependen the:(G)

number, clique the: (G)

(G), (G) (G)

|V(G)| (G), Max

G,any For

cf

.(G)

|V(G)| (G)

then ,transitive- vertexisG If

f

Density of Sequences w/ Missing DifferencesLet D be a set of positive integers.

Example, D = {1, 4, 5}.

“density” of this M(D) is 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 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 [Chang, L., Zhu, 1999]

For any finite set of integers D,

,D)) (G(n,

n

(D)

1 )),(( lim

n f

DZG

where G(n, D) is the subgraph induced by {0, 1, 2, …, n-1}.

Lonely Runner Conjecture

Suppose k runners running on a circular field of circumference 1. 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 1/k apart from all other runners.

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

Suppose there are k runners

Fix one runner at the same origin point with speed 0. For other runners, take relative speeds to this fixed runner. Hence we get |D| = k – 1.

For example, two runners, then D = { d }

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 D}.

(D) : = sup { || D t ||: t 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

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

Wills Conjecture [1967]

For any D, 1 |D|

1 (D)

Bienia et al, View obstruction and the lonely runner,

1998). Another proof for 5 runners. Y.-G. Chen, J. Number Theory, 1990 &1991.

(A more generalized conjecture.)

Wills, Diophantine approximation, 1967. Betke and Wills, 1972. (Proved for 4 runners.) Cusick and Pomerance, 1984. (Proved for 5 runners.)

The conjecture is confirmed for:

7 runners (Barajas and Serra, 2007)

5 runners [Cussick and Pomerance, 1984]

[Bienia et al., 1998]

6 runners [Holzman and Kleitman, 2001]

Graph homorphism

For two graphs G and H, graph homomorphism is a function V(G) → V(H) such that if u ~G v then f(u) ~H f(H).

If such a function exists, denote G → H.

Circular cliques and circular chromatic number

For given positive integers p ≥ 2q, the circular clique Kp/q has vertex set V = {0, 1, 2, …, p - 1} u ~ v iff |u – v|p ≥ q

χ c (G) ≤ p/q iff G → Kp/q

Circulant graphs and distance graphs

For a positive integer n and a set D of a positive integers with n ≥ 2Max {D}. The circulant graph generated by D with order n, denoted by G(Z n,D), has V = {0, 1, 2, . . . , n – 1} u ~ v iff |u – v| D or n - |u – v| D.

G (Z, D) → G(Z n, D) for all n ≥ 2Max {D}.

Hence, χc (G (Z, D)) ≤ χc (G(Z n, D)).

Relations

1 |D| (D)

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

f

?

(D)

1

| |

Lonely Runner Conjecture

Zhu, 2001

Chang, L., Zhu, 1999

More than ten papers…

D = {a, b}

Note, always assume gcd (D) = 1.

If a, b are odd, then G(Z, D) is bipartite, and (D) = (D) = ½.

If a, b are of different parity, then

(D) = (D) = (a+b-1)/2(a+b).

Almost Difference Closed Sets

Definition: Sets D with (G(Z, D)) = |D|.Theorem [L & Zhu, 2004]: Let gcd(D)=1.

(G(Z, D)) = |D| iff D is one of:

A.1. D = { 1, 2, …, a, b }

A.3. D = { x, y, y – x, y + x }, y > x, y 2x.

A.2. D = { a, b, a + b }

(D) = (D)

(D) = (D)

(D) solved, (D) partially open

Theorem & Conjecture [L & Zhu, 2004]

Theorem: If D = { a, b, a + b }, gcd(a, b, c)=1, then

} a2b

3a2b

,b2a

3b2a

{Max (D)

[Conjectured 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} = 4/13

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

1991][Chen (D)

Conjecture [L. & Zhu, 2004]

If D = {x, y, y - x, y + x} where x = 2k+1 and

y = 2m + 1, m > k, then

? 1m 1)(k 4

m 1)(k )(

D

Example: μ ({2, 3, 5, 8}) = ?

Punched Sets Dm,k,s = [m] - {k, 2k, …, sk}

When s = 1.

When s > 1.

[Eggleton et al., 1985] Some χ(G)[Kemnitz and Kolberg, 1998] Some χ (G)[Chang et al., 1999] Completely solved χf (G), χ(G).[Chang, Huang and Zhu, 1998] Completed χc (G).

[L. & Zhu, 1999] Completed χf (G) and χ (G). [Huang and Chang, 2000] Found D, χc(G) < 1/(D)[Zhu, 2003] Completed χc (G).

Unions of Two Intervals

Dm, [a,b] = [1, m] – [a, b] = [1, a-1] [b+1, m].

[Wu and Lin, 2004] Complete χf (G) for b < 2a

[Lam, Lin and Song, 2005] Completed χ (G) and partially χc (G), for b < 2a.

[Lam and Lin, 2005] Partially χf (G) for b 2a.

[L. and Zhu, 2008] Completed χf (G) for all a, b, m.

For χc (G) in general, Open problem.

Open Problem and Conjecture

Conjecture [Zhu, 2002]:

If (G(Z, D)) < |D| then χ (G(Z, D)) ≤ |D|.

|D| = 3 [Zhu, 2002]

|D| = 4 [Barajas and Serra, 2007]

|D| > 4, open. ?