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(R 2, {1}) = Unit Distance Graph of R 2. χ ( G(R 2, {1})) = χ ( R 2, {1}) = ? 4 ≤ χ ( R 2, {1}) ≤ 7 [Moser & Moser, 1968; Hadweiger et al., 1964]

< 1

At least we need four colors for coloring the plane Assume only use three colors: red, blue and green. X 1

Rational Points on the Plane [van Luijk, Beukers, Israel, 2001]

Distance Graphs (Eggleton, Erdős, Skelton ) 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} 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 χ f (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

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 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, 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, 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, Betke and Wills, (Proved for 4 runners.) Cusick and Pomerance, (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 K p/q has vertex set  V = {0, 1, 2, …, p - 1}  u ~ v iff |u – v| p ≥ q χ c (G) ≤ p/q iff G → K p/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, χ c (G (Z, D)) ≤ χ c (G(Z n, D)).

Relations ? 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) solved,  (D) partially open

Theorem & Conjecture [L & Zhu, 2004] Theorem: If D = { a, b, a + b }, gcd(a, b, c)=1, then [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,....

Conjecture [L. & Zhu, 2004] If D = {x, y, y - x, y + x} where x = 2k+1 and y = 2m + 1, m > k, then Example: μ ({2, 3, 5, 8}) = ?

Punched Sets D m,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 χ f (G), χ(G). [Chang, Huang and Zhu, 1998]  Completed χ c (G). [L. & Zhu, 1999]  Completed χ f (G) and χ (G). [Huang and Chang, 2000]  Found D, χ c (G) < 1/  (D) [Zhu, 2003]  Completed χ c (G).

Unions of Two Intervals D m, [a,b] = [1, m] – [a, b] = [1, a-1]  [b+1, m].  [Wu and Lin, 2004]  Complete χ f (G) for b < 2a  [Lam, Lin and Song, 2005]  Completed χ (G) and partially χ c (G), for b < 2a.  [Lam and Lin, 2005]  Partially χ f (G) for b  2a.  [L. and Zhu, 2008]  Completed χ f (G) for all a, b, m.  For χ c (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. ?