Slides - Iowa State University

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On the independence ratio of distance graphs
James M. Carraher
David Galvin Stephen G. Hartke
Derrick Stolee∗
A. J. Radcliffe
Iowa State University
[email protected]
http://www.math.iastate.edu/dstolee/r/distance.htm
January 18th, 2014
AMS/MAA Joint Math Meetings
James M. Carraher
University of Nebraska–Lincoln
[email protected]
David Galvin
University of Notre Dame
[email protected]
Stephen G. Hartke
[email protected]
A. J. Radcliffe
[email protected]
Distance Graphs
For a set S of positive integers, the distance graph G (S ) is the infinite graph with
vertex set Z where
two integers i and j are adjacent if and only if |i − j | ∈ S.
Circulant Graphs
For an integer n, the circulant graph G (n, S ) is the graph whose vertices are the
integers modulo n where
two integers i and j are adjacent if and only if |i − j | ≡ k (mod n), for some k ∈ S.
-5
-6
5
-4
4
-3
3
-2
2
-1
0
1
G (12, {1, 4, 5})
-6
6
-5
5
-4
4
-3
3
-2
2
-1
0
1
G (18, {1, 4, 5})
-6
6
-5
5
-4
4
-3
3
-2
2
-1
0
1
G (28, {1, 4, 5})
-6
6
-5
5
-4
4
-3
3
-2
2
-1
0
1
G (40, {1, 4, 5})
-5
5
-4
4
-3
3
-2
-1
0
1
G (60, {1, 4, 5})
2
-5
5
-4
4
-3
3
-2
-1
0
1
G (128, {1, 4, 5})
2
-4
-3
-2
-1
0
1
G (256, {1, 4, 5})
2
3
4
-4
-3
-2
-1
0
1
G (512, {1, 4, 5})
2
3
4
-4
-3
-2
-1
0
1
G (1024, {1, 4, 5})
2
3
4
-4
-3
-2
-1
0
1
G (2048, {1, 4, 5})
2
3
4
-4
-3
-2
-1
0
1
G (∞, {1, 4, 5}) = G ({1, 4, 5})
2
3
4
Chromatic Number of Distance Graphs
J. Barahas, O. Serra, Distance graphs with maximum chromatic number, Discrete Mathematics 308 (2008)
1355–1365.
G. J. Chang, D. D.-F. Liu, X. Zhu, Distance Graphs and T -Coloring, Journal of Comb. Theory, Ser. B. 75 (1999)
259–269.
J.-J. Chen, G. J. Chang, K.-C. Huang, Integral Distance Graphs, Journal of Graph Theory 25 (1997) 287–294.
W. A. Deuber, X. Zhu, The chromatic numbers of distance graphs, Discrete Mathematics 165 (1997) 194–204.
R.B. Eggleton, P. Erdős, D.K. Skilton, Colouring the real line, J. Combin. Theory B 39 (1985) 86–100.
R.B. Eggleton, P. Erdős, D.K. Skilton, Coloring prime distance graphs, Graphs Combin. 32 (1990) 17–32.
C. Heuberger, On planarity and colorability of circulant graphs, Discrete Mathematics 268 (2003) 153–169.
A. Ilić, M. Basić, On the chromatic number of integral circulant graphs, Computers and Mathematics with
Applications 60 (2010) 144–150.
Y. Katznelson, Chromatic numbers of Cayley graphs on Z and recurrence, Combinatorica 21(2) (2001) 211–219.
A. Kemnitz, H. Kolberg, Coloring of integer distance graphs, Discrete Mathematics 191 (1998) 113–123.
P. C. B. Lam, W. Lin, Coloring of distance graphs with intervals as distance sets, European Journal of
Combinatorics 26 (2005) 1216–1229.
D. D.-F. Liu, From rainbow to the lonely runner: A survey on coloring parameters of distance graphs, Taiwanese
Journal of Mathematics 12 (2008) 851–871.
D. D.-F. Liu, X. Zhu, Distance graphs with missing multiples in the distance sets, Journal of Graph Theory 30
(1999) 245–259.
I. Z. Ruzsa, Zs. Tuza, M. Voigt, Distance Graphs with Finite Chromatic Number, Journal of Combinatorial Theory,
Series B 85 (2002) 181–187.
Fractional Chromatic Number
A fractional coloring of a graph G is a feasible solution to the following linear
program:
min ∑I ∈I cI
∑ I 3 v cI
cI
≥ 1 ∀v ∈ V (G )
≥ 0 ∀I ∈ I
where I is the collection of independent sets in G.
Fractional Chromatic Number
A fractional coloring of a graph G is a feasible solution to the following linear
program:
min ∑I ∈I cI
∑ I 3 v cI
cI
≥ 1 ∀v ∈ V (G )
≥ 0 ∀I ∈ I
where I is the collection of independent sets in G.
The fractional chromatic number χf (G ) is the minimum value of a fractional
coloring, and provides a lower bound on the chromatic number.
Fractional Chromatic Number of Distance Graphs
J. Brown, R. Hoshino, Proof of a conjecture on fractional Ramsey numbers, Journal of Graph Theory 63(2)
(2010) 164–178.
G. H. Chang, L. Huang, and X. Zhu, Circular chromatic numbers and fractional chromatic numbers of distance
graphs, European J. of Combinatorics 19 (1998) 423–431.
K. L. Collins, Circulants and sequences, SIAM J. Discrete Math. 11 (1998) 330–339.
G. Gao and X. Zhu, Star extremal graphs and the lexicographic product, Discrete Math. 165/166 (1996) 147–156.
R. Hoshino, Independence polynomials of circulant graphs, Ph.D. Thesis, Dalhousie University (2007).
K.-W. Lih, D. D.-F. Liu, X. Zhu, Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491–499.
X. Zhu, Circular Chromatic Number of Distance Graphs with Distance Sets of Cardinality 3, Journal of Graph
Theory 41 (2002) 195–207.
Independence Ratio
For an independent set A in G (S ) the density δ(A) is equal to
δ(A) = lim sup
N →∞
|A ∩ [−N, N ]|
.
2N + 1
Independence Ratio
δ(A) = lim sup
N →∞
|A ∩ [−N, N ]|
.
2N + 1
The independence ratio α(S ) is the supremum of δ(A) over all independent sets
A in G (S ).
Independence Ratio
δ(A) = lim sup
N →∞
|A ∩ [−N, N ]|
.
2N + 1
The independence ratio α(S ) is the supremum of δ(A) over all independent sets
A in G (S ).
Theorem (Lih, Liu, and Zhu, ’99) Let S be a finite set of positive integers.
χf (G (S )) =
1
.
α (S )
Periodic Independent Sets to Fractional Colorings
Suppose X ⊂ Z is a periodic independent set in G (S ) with period p and density
δ(X ) = dp .
δ(X ) = dp .
Let Xi = X + i for i ∈ {0, . . . , p − 1}.
Assign cXi = d1 .
δ(X ) = dp .
Let Xi = X + i for i ∈ {0, . . . , p − 1}.
Assign cXi = d1 . Every vertex appears in d sets Xi , so d ·
1
d
≥ 1.
δ(X ) = dp .
Let Xi = X + i for i ∈ {0, . . . , p − 1}.
Assign cXi = d1 . Every vertex appears in d sets Xi , so d ·
1
d
≥ 1.
Then
χf (G (S )) ≤ ∑pi =−01 cXi =
p
d
=
1
.
δ (X )
Periodic Independent Sets
Theorem (CGHRS, ’14+) Let S be a finite set of positive integers and let
s = max S.
There exists a periodic independent set A in G (S ) with period at most s2s where
δ (A) = α (S ).
Cycle Lemma
Let G be a finite digraph with weights on the vertices.
Cycle Lemma
An infinite walk in G is a sequence W = (wi )i ∈Z such that wi wi +1 is an edge for
all i ∈ Z.
Cycle Lemma
all i ∈ Z.
The (upper) average weight of W is w (W ) = lim supN →∞
∑N
i =−N wi
2N +1 .
Cycle Lemma
all i ∈ Z.
The (upper) average weight of W is w (W ) = lim supN →∞
∑N
i =−N wi
2N +1 .
Lemma (Cycle Lemma) Let G be a finite, vertex-weighted digraph. The
supremum of upper average weights of infinite walks on G is equal to the upper
average weight of some infinite walk given by repeating a simple cycle.
Proof of Cycle Lemma
Let W = (xi )i ∈Z be an infinite walk in G and let N n(G ).
Let W = (xi )i ∈Z be an infinite walk in G and let N n(G ). Let WN = (xi )N
i =−N
N
w
(
x
)
∑
i
and observe that for large N, i =−N
closely approximates w (W ).
2N + 1
i =−N
and observe that for large N,
∑N
i =−N w (xi )
2N +1
∑N
i =−N w (xi )
is closely approximated by a
2N + 1
convex combination of w (C1 ), . . . , w (Ct ).
There exist cycles C1 , . . . , Ct such that
i =−N
and observe that for large N,
∑N
i =−N w (xi )
2N +1
∑N
i =−N w (xi )
is closely approximated by a
2N + 1
convex combination of w (C1 ), . . . , w (Ct ).
There exist cycles C1 , . . . , Ct such that
Thus, in the limit, w (W ) ≤ max{ w (C ) : C is a cycle in G }.
Proof of Periodic Sets
A state σ is a subset of {0, . . . , s − 1} (there are 2s such states).
A state σ is a subset of {0, . . . , s − 1} (there are 2s such states).
A state is allowed if σ is independent in G (S ).
The state diagram of allowed states is a digraph where an ordered pair (σ1 , σ2 ) of
states be an edge if and only if σ1 ∪ (s + σ2 ) is independent in G (S ).
The independent sets X in G (S ) are in bijection with the infinite walks W in the
state diagram, and the density of X equals the average weight of its
corresponding walk, WX .
By the cycle lemma, the maximum density of an infinite walk in the state diagram
is achieved by repeating a simple cycle C.
is achieved by repeating a simple cycle C. The independent set XC given by those
states has maximum density δ(XC ) = α(S ).
is achieved by repeating a simple cycle C. The independent set XC given by those
states has maximum density δ(XC ) = α(S ).
The length of C is at most 2s , so the period of XC is at most s2s .
Other Periodic Sets
Let S be a finite set of positive integers and set s = max S.
Theorem (CGHRS, ’14+) The minimum density of a dominating set in G (S ) is
achieved by a periodic set with period at most (2s )22s .
Theorem (CGHRS, ’14+) The minimum density of a 1-identifying code in G (S )
is achieved by a periodic set with period at most (6s )26s .
Corollary (CGHRS, ’14+) The minimum density of an r -identifying code in G (S )
is achieved by a periodic set with period at most (6sr )26sr .
Theorem (Eggleton, Erdős, and Skilton, ’90) For k = χ(G (S )), there exists a
periodic proper k -coloring c with minimum period at most sk s .
Example Theorem (S = {1, 2, k })
Theorem (Zhu, ’02)
α({1, 2, k }) =

k


3k +3


1
3



1
3
if k ≡ 0
(mod 3)
if k ≡ 1
(mod 3)
if k ≡ 2
(mod 3).
Theorem (CGHRS, ’14+) For k > 4,

2k


5k +5




2



5
+1
α({1, 4, k }) = 2k
5k +5




2k −1


 5k +5



2
5
if k ≡ 0
(mod 5)
if k ≡ 1
(mod 5)
if k ≡ 2
(mod 5)
if k ≡ 3
(mod 5)
if k ≡ 4
(mod 5).
0.441 mod 5
−→
0.39
0.38
0.37
2 mod 5
0.36
α({1, 4, k }) =
0.35
 2k


5k +5



2



5
if k ≡ 0 (mod 5)



2k −1

 5k +5



2
if k ≡ 3 (mod 5)
if k ≡ 1 (mod 5)
2k +1
 5k +5
if k ≡ 2 (mod 5)
if k ≡ 4 (mod 5).
5
0.34
30 mod 5
0.33
0
20
40
60
80
100
2
5
Discharging
Suppose X = {xi : i ∈ Z} ⊆ Z is an infinite independent set in G (S ).
Discharging
Elements are labeled . . . , x−1 , x0 , x1 , . . . , xi , . . . .
Discharging
Blocks are sets Bk = {xk , xk + 1, . . . , xk +1 − 1}.
(“Intervals” closed on element xk and open on xk +1 )
Discharging
Frames are collections Fj = {Bj , Bj +1 , . . . , Bj +t −1 }.
(There are t blocks in each frame.)
Local Discharging Lemma
Let a, b, c, t be nonnegative integers. Let X be a periodic independent set in G (S ).
Stage 1: Blocks
µ(Bj ) = a|Bj | − b
discharge
/ µ ∗ ( Bj )
Stage 2: Frames
defines
ν ∗ ( Fj )
discharge
/ ν0 (Fj ) ≥ c
Stage 1: Blocks
µ(Bj ) = a|Bj | − b
discharge
/ µ ∗ ( Bj )
defines
ν ∗ ( Fj )
Stage 2: Frames
If ν0 (Fj ) ≥ c for all frames, then
δ (X ) ≤
at
.
bt + c
discharge
/ ν0 (Fj ) ≥ c
Example Discharging Argument (S = {1, 4, k })
Theorem (CGHRS, ’14+) For k > 4,

2k


5k +5


2


5
+1
α({1, 4, k }) = 2k
5k +5


2k −1



5k +5

2
5
if k
if k
if k
if k
if k
≡0
≡1
≡2
≡3
≡4
(mod
(mod
(mod
(mod
(mod
5)
5)
5)
5)
5).
Always, let a = 2 and b = 5.
Residue class
t
c
Extremal Set
k =5i
t = 2i
c=2
(2 3)i −1 32
k =5i + 1
t =1
c=0
23
k =5i + 2
t = 2i + 1
c=1
(2 3)i 3
k =5i + 3
t = 2i + 1
c=3
(2 3)i −1 33
k =5i + 4
t =1
c=0
23
Block Size
µ-charge
µ∗ -charge
2
−1
0
3
1
0, 1
5
5
4, 5
6
7
6, 7
For all cases, let a = 2 and b = 5.
Residue class
t
c
Extremal Set
k =5i
t = 2i
c=2
(2 3)i −1 32
k =5i + 1
t =1
c=0
23
k =5i + 2
t = 2i + 1
c=1
(2 3)i 3
k =5i + 3
t = 2i + 1
c=3
(2 3)i −1 33
k =5i + 4
t =1
c=0
23
Block Size
µ-charge
µ∗ -charge
2
−1
0
3
1
0, 1
5
5
4, 5
6
7
6, 7
(S1) Every 2-block Bj pulls one unit of charge from Bj +1 .
+1
Case k = 5i + 3:
Let t = 2i + 1 and c = 3. Thus
at
bt +c
=
4i +2
10i +8
=
2i +1
5i +4
=
2k −1
5k +5 .
Case k = 5i + 3:
second-stage discharging rule.
at
bt +c
=
4i +2
10i +8
=
2i +1
5i +4
=
2k −1
5k +5 .
We use the following
(S2) If σ(Fj ) = ∑B` ∈Fj |B` | = 5i + 2, then Fj pulls 1 unit of charge from each of
Fj +1 , Fj +2 , and Fj +3 .
Case k = 5i + 3:
at
bt +c
=
4i +2
10i +8
=
2i +1
5i +4
=
2k −1
5k +5 .
Fj +1 , Fj +2 , and Fj +3 .
It remains to show that:
1. if ν∗ (Fj ) < 3 = c, then σ(Fj ) = 5i + 2 and Fj pulls charge by (S2).
Case k = 5i + 3:
at
bt +c
=
4i +2
10i +8
=
2i +1
5i +4
=
2k −1
5k +5 .
Fj +1 , Fj +2 , and Fj +3 .
It remains to show that:
1. if ν∗ (Fj ) < 3 = c, then σ(Fj ) = 5i + 2 and Fj pulls charge by (S2).
2. if Fj loses charge by (S2), then Fj contains a 5-block and ν∗ (Fj ) ≥ 4.
Densities for S = {1, 1 + k , 1 + k + i }
Future Work
Goal: Characterize α({i, j, k }) for all 1 ≤ i < j < k . (or just i = 1?)
Discharging arguments for |S | > 3?
Stronger bounds on minimum period?
On the independence ratio of distance graphs
James M. Carraher
David Galvin Stephen G. Hartke
Derrick Stolee∗
A. J. Radcliffe
Iowa State University
[email protected]
http://www.math.iastate.edu/dstolee/r/distance.htm
January 18th, 2014
AMS/MAA Joint Math Meetings

Slides - Iowa State University

Transcription

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