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Coloring Parameters of Distance Graphs

Coloring Parameters of Distance Graphs. Daphne Liu Department of Mathematics California State Univ., Los Angeles. Overview:. Plane coloring. Fractional Chromatic Number. Lonely Runner Conjecture. Distance Graphs. Circular Chromatic Number. Plane Coloring Problem.

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Coloring Parameters of Distance Graphs

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

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

  3. 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]

  4. < 1

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

  6. Rational Points on the Plane [van Luijk, Beukers, Israel, 2001] http://www.math.leidenuniv.nl/~naw/serie5/deel01/sep2000/pdf/problemen3.pdf

  7. 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.

  8. Example D = {1, 3, 4} 1 2 3 5 6 7 8 4 0 Note: For any D, χ (G(Z, D)) ≤ |D| + 1.

  9. Chromatic number of G(Z, P) • D = P, set of all primes. Then χ(G(Z, P)) = 4.[Eggleton et. al. 1985] • Open Problem: For what D  P, χ(G(Z, D)) = 4 ? This problem is solved for |D| = 3, 4. [Eggleton et al 1985] [Voigt and Walther 1994]

  10. 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.

  11. Facts on Fractional Chromatic Number

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

  13. 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}.

  14. 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.

  15. 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 }

  16. 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}.

  17. 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.

  18. Wills Conjecture [1967] For any D, • Wills, Diophantine approximation, 1967. • Betke and Wills, 1972. (Proved for 4 runners.) • Cusick and Pomerance, 1984. (Proved for 5 runners.) • 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.)

  19. 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]

  20. 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.

  21. 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 • χc (G) ≤ p/q iff G → Kp/q

  22. 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)).

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

  24. 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).

  25. 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 }  (D) =  (D) A.2. D = { a, b, a + b }  (D) =  (D) A.3. D = { x, y, y – x, y + x }, y > x, y  2x.   (D) solved, (D)partially open

  26. 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 ({1, 4, 5}) = Max { 1/3, 1/3} = 4/13 Example: μ ({3, 5, 8}) = Max { 2/11, 4/13} = 4/13 M(D) = 0, 2, 4, 6, 13, 15, 17, 19, 26, . . . .

  27. 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}) = ?

  28. Punched Sets Dm,k,s = [m] - {k, 2k, …, sk} • 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). • When s > 1. [L. & Zhu, 1999]  Completedχf (G) and χ (G). [Huang and Chang, 2000]  Found D, χc(G) < 1/(D) [Zhu, 2003]  Completedχc (G).

  29. 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]  Partiallyχf (G) for b  2a. • [L. and Zhu, 2008]  Completed χf (G) for all a, b, m. • For χc (G) in general, Open problem.

  30. 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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