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Coloring Parameters of Distance Graphs Daphne Liu Department of Mathematics California State Univ., Los Angeles

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Overview: Distance Graphs Fractional Chromatic Number Lonely Runner Conjecture Plane coloring Circular Chromatic Number

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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]

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< 1

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

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

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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.

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D = {1, 3, 4} Example Note: For any D, χ (G(Z, D)) ≤ |D| + 1.

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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 ?

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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.

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Facts on Fractional Chromatic Number

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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.

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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}.

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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.

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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 }

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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}.

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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.

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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.)

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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]

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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.

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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

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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)).

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Relations ? Lonely Runner Conjecture Zhu, 2001 Chang, L., Zhu, 1999 More than ten papers…

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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).

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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

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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,....

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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}) = ?

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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).

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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.

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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. ?

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