coloring parameters of distance graphs
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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. - PowerPoint PPT PresentationTRANSCRIPT
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Coloring Parameters of Distance Graphs
Daphne LiuDepartment 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(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]
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< 1
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At least we need four colors for coloring the planeAssume only use three colors: red, blue and green.
X
1
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Rational Points on the Plane
2 ){1} ,(Q 2
http://www.math.leidenuniv.nl/~naw/serie5/deel01/sep2000/pdf/problemen3.pdf
[van Luijk, Beukers, Israel, 2001]
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Distance Graphs (Eggleton, Erdős, Skelton 1985 - 1987)
Defined on the real line: Given a set D ofpositive 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 ofpositive 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}
0 1 2 3 4 5 6 7 8
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 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.
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Facts on Fractional Chromatic Number
number. ceindependen the:(G) number, clique the: (G)
(G), (G) (G)
|V(G)| (G), Max
G,any For
cf
.(G)
|V(G)| (G)
then ,transitive- vertexisG If
f
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Density of Sequences w/ Missing DifferencesLet 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,
,D)) (G(n,
n (D) 1 )),(( lim
n f
DZG
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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ExampleD = {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, 1 |D|
1 (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, 1967. Betke and Wills, 1972. (Proved for 4 runners.) Cusick and Pomerance, 1984. (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 Kp/q has vertex set V = {0, 1, 2, …, p - 1} u ~ v iff |u – v|p ≥ q
χ c (G) ≤ p/q iff G → Kp/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, χc (G (Z, D)) ≤ χc (G(Z n, D)).
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Relations
1 |D| (D)1 D)) (G(Z, D)) (G(Z, c
f
?
(D) 1| |
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) = (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
} a2b
3a2b
,b2a
3b2a
{Max (D)
[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, . . . .
1991][Chen (D)
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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
? 1m 1)(k 4
m 1)(k )(
D
Example: μ ({2, 3, 5, 8}) = ?
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Punched Sets Dm,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 χf (G), χ(G).[Chang, Huang and Zhu, 1998] Completed χc (G).
[L. & Zhu, 1999] Completed χf (G) and χ (G). [Huang and Chang, 2000] Found D, χc(G) < 1/(D)[Zhu, 2003] Completed χc (G).
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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] Partially χf (G) for b 2a.
[L. and Zhu, 2008] Completed χf (G) for all a, b, m.
For χc (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. ?