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: D = { 1, 3, 4 }

Example D = {1, 3, 5, 7, 9}. Then χ (D) = 2

3 – Element Sets Let D = { a, b, c }, a < b < c. [Eggleton, Erdős, Skilton 1985] [Chen, Huang, Chang 1997][Voigt, 1999] [Zhu, 2002]

Chromatic Number of Distance Graphs If D is a subset of prime numbers then χ (D) ≤ 4. If D contains only odd numbers then χ (D) ≤ 2. If D contains no multiples of k, then χ (D) ≤ k.

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). [Voigt and Walther, 1994] [Eggleton, Erdős, Skilton 1990]

General 4 – element Sets D = { 2, 3, x, x+s }, x > 3. Voigt and Walther proved that χ (D) = 3 if s ≥ 10 and x ≥ s 2 – 6s +3. Kemnitz and Kolberg determined the chromatic number for all s < 10. Our aim: Completely solve this problem.

Theorems IfD = {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, 1999Zhu, 2001 Let D = {2, 3, x, y}. Then

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 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, 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 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 For any D, κ (D) = m/n, where n is the sum of two elements in D.

Idea 0 1/3

Theorem [L. and Sutedja 2011] D = { 2, 3, x, x+10 }. D = { 2, 3, 6, y }. D = { 2, 3, 10, y } D = { 2, 3, 4, y }.

Alternative Definition of κ (D)

Theorem [L. and Sutedja 2011] Let D ={ 2, 3, x, y }. Then χ (D)=3 for: X = 9, y ≥ 36 X = 11, y ≥ 66 X = 15, y ≥ 60 X = 16, y ≥ 48 X ≥ 53, y ≥ x - 11 X = 12, 13, 14, or x ≥ 17, and y ≥ 2 x

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 For any D, Wills, Diophantine approximation, in German, Cusick and Pomerance, (True for |D| ≤ 4.) Chen, J. Number Theory, a generalized conjecture. Bohman, Holzman and Kleitman |D| = 5. Barajas & Serra |D| = 6.