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From Rainbow to the Lonely Runner Daphne Liu Department of Mathematics California State Univ. 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 Overview: Plane coloring Distance Graphs Fractional Chromatic Number Circular Chromatic Number Lonely Runner Conjecture

Plane Coloring Problem l Color all the points on the xy-plane so that any Plane Coloring Problem l Color all the points on the xy-plane so that any two points of unit distance apart get different colors. l What is the smallest number of colors needed to accomplish the above ? l Seven colors are enough [Moser & Moser, 1968]

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Graphs and Chromatic Number l A graph G contains two parts: Vertices and edges. Graphs and Chromatic Number l A graph G contains two parts: Vertices and edges. l 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. l

Example Example

At least we need four colors for coloring the plane Assume three colors, red, At least we need four colors for coloring the plane Assume three colors, red, blue and green, are used. X

Known Facts [Moser & Moser, 1968; Hadweiger et al. , 1964] [van Luijk, Beukers, Known Facts [Moser & Moser, 1968; Hadweiger et al. , 1964] [van Luijk, Beukers, Israel, 2001] http: //www. math. leidenuniv. nl/~naw/serie 5/deel 01/sep 2000/pdf/problemen 3. pdf

Circular Chromatic Number l l l Let G be a graph. Let r be Circular Chromatic Number l l l 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.

Example, C 5 0 0 1 1. 5 0. 5 2 1. 5 1 Example, C 5 0 0 1 1. 5 0. 5 2 1. 5 1

Known Results: The following hold for any graph G: → 2 Known Results: The following hold for any graph G: → 2

Distance Graphs Eggleton, Erdős et. al. [1985 – 1987] l For a given set Distance Graphs Eggleton, Erdős et. al. [1985 – 1987] l 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 l l Suppose k runners running on a circular field of Lonely Runner Conjecture l l 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 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 l D = {1, 3, 4} (Four runners) ||(1/3) D|| = min {1/3, Example l D = {1, 3, 4} (Four runners) ||(1/3) D|| = min {1/3, 0, 1/3} = 0 ||(1/4) D|| = min {1/4, 0} = 0 ||(1/7) D|| = min {1/7, 3/7} = 1/7 ||(2/7) D|| = min {2/7, 1/7} = 1/7 ||(3/7) D|| = min {3/7, 2/7} = 2/7 φ (D) = 2/7 [Chen, J. Number Theory, 1991] ≥ ¼.

Wills Conjecture For any D, l l l Wills, Diophantine approximation, in German, 1967. Wills Conjecture For any D, l l l 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 ? Lonely Runner Conjecture Relations L. & Zhu, J. Graph Theory, 2004 Zhu, 2001 ? Lonely Runner Conjecture Chang, L. , Zhu, 1999

Density of Sequences w/ Missing Differences l Let D be a set of positive Density of Sequences w/ Missing Differences l Let D be a set of positive integers. Example, D = {1, 4, 5}. => μ ({1, 4, 5}) = 1/3. l 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. l μ (D) = maximum density of an M(D).

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