James M. Carraher David Galvin Stephen G. Hartke A. J. Radcliffe …orion.math.iastate.edu/dstolee/presentations/2014-01-JMM.pdf · 2014-01-18 · James M. Carraher David Galvin University

On the independence ratio of distance graphs

James M. Carraher David Galvin Stephen G. Hartke A. J. RadcliffeDerrick Stolee∗

Iowa State [email protected]

http://www.math.iastate.edu/dstolee/r/distance.htm

January 18th, 2014AMS/MAA Joint Math Meetings

James M. Carraher David GalvinUniversity of Nebraska–Lincoln University of Notre [email protected] [email protected]

Stephen G. Hartke A. J. RadcliffeUniversity of Nebraska–Lincoln University of Nebraska–[email protected] [email protected]

Distance Graphs

For a set S of positive integers, the distance graph G(S) is the infinite graph withvertex 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 theintegers modulo n where

two integers i and j are adjacent if and only if |i − j | ≡ k (mod n), for some k ∈ S.

0-1 1

-2 2

-3 3

-4 4

-5 5-6

G(12, {1,4,5})

0-1 1

-2 2

-3 3

-4 4

-5 5

-6 6

G(18, {1,4,5})

0-1 1

-2 2

-3 3

-4 4

-5 5

-6 6

G(28, {1,4,5})

0-1 1

-2 2

-3 3

-4 4

-5 5

-6 6

G(40, {1,4,5})

0-1 1

-2 2

-3 3

-4 4

-5 5

G(60, {1,4,5})

0-1 1-2 2

-3 3

-4 4

-5 5

G(128, {1,4,5})

0-1 1-2 2

-3 3

-4 4

G(256, {1,4,5})

0-1 1-2 2-3 3

-4 4

G(512, {1,4,5})

0-1 1-2 2-3 3-4 4

G(1024, {1,4,5})

0-1 1-2 2-3 3-4 4

G(2048, {1,4,5})

-4 -3 -2 -1 0 1 2 3 4

G(∞, {1,4,5}) = G({1,4,5})

Chromatic Number of Distance GraphsJ. 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. Erdos, D.K. Skilton, Colouring the real line, J. Combin. Theory B 39 (1985) 86–100.

R.B. Eggleton, P. Erdos, 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. Ilic, M. Basic, On the chromatic number of integral circulant graphs, Computers and Mathematics withApplications 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 ofCombinatorics 26 (2005) 1216–1229.

D. D.-F. Liu, From rainbow to the lonely runner: A survey on coloring parameters of distance graphs, TaiwaneseJournal 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 linearprogram:

min ∑I∈I cI

∑I3v cI ≥ 1 ∀v ∈ V (G)cI ≥ 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 fractionalcoloring, and provides a lower bound on the chromatic number.

Fractional Chromatic Number

A fractional coloring of a graph G is a feasible solution to the following linearprogram:

min ∑I∈I cI

∑I3v cI ≥ 1 ∀v ∈ V (G)cI ≥ 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 fractionalcoloring, 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 distancegraphs, 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 GraphTheory 41 (2002) 195–207.

Independence Ratio

For an independent set A in G(S) the density δ(A) is equal to

δ(A) = lim supN→∞

|A∩ [−N,N ]|2N + 1

.

The independence ratio α(S) is the supremum of δ(A) over all independent setsA in G(S).

Theorem (Lih, Liu, and Zhu, ’99) Let S be a finite set of positive integers.

χf (G(S)) =1

α(S).

Independence Ratio



|A∩ [−N,N ]|2N + 1

.



χf (G(S)) =1

α(S).

Independence Ratio



|A∩ [−N,N ]|2N + 1

.



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 .

Let Xi = X + i for i ∈ {0, . . . ,p− 1}.

Assign cXi =1d . Every vertex appears in d sets Xi , so d · 1

d ≥ 1.

Then

χf (G(S)) ≤ ∑p−1i=0 cXi =

pd =

1δ(X )

.



δ(X ) = dp .

Let Xi = X + i for i ∈ {0, . . . ,p− 1}.

Assign cXi =1d .

Every vertex appears in d sets Xi , so d · 1d ≥ 1.

Then

χf (G(S)) ≤ ∑p−1i=0 cXi =

pd =

1δ(X )

.



δ(X ) = dp .

Let Xi = X + i for i ∈ {0, . . . ,p− 1}.


d ≥ 1.

Then

χf (G(S)) ≤ ∑p−1i=0 cXi =

pd =

1δ(X )

.



δ(X ) = dp .

Let Xi = X + i for i ∈ {0, . . . ,p− 1}.


d ≥ 1.

Then

χf (G(S)) ≤ ∑p−1i=0 cXi =

pd =

1δ(X )

.

Periodic Independent Sets

Theorem (CGHRS, ’14+) Let S be a finite set of positive integers and lets = max S.

There exists a periodic independent set A in G(S) with period at most s2s whereδ(A) = α(S).

Cycle Lemma

Let G be a finite digraph with weights on the vertices.

An infinite walk in G is a sequence W = (wi)i∈Z such that wiwi+1 is an edge forall i ∈ Z.

The (upper) average weight of W is w(W ) = lim supN→∞∑N

i=−N wi2N+1 .

Lemma (Cycle Lemma) Let G be a finite, vertex-weighted digraph. Thesupremum of upper average weights of infinite walks on G is equal to the upperaverage weight of some infinite walk given by repeating a simple cycle.

Cycle Lemma




i=−N wi2N+1 .


Cycle Lemma




i=−N wi2N+1 .


Cycle Lemma




i=−N wi2N+1 .


Proof of Cycle Lemma

Let W = (xi)i∈Z be an infinite walk in G and let N � n(G).

Let WN = (xi)Ni=−N

and observe that for large N,∑N

i=−N w(xi)

2N + 1closely approximates w(W ).


Let W = (xi)i∈Z be an infinite walk in G and let N � n(G). Let WN = (xi)Ni=−N

and observe that for large N,∑N

i=−N w(xi)

2N + 1closely approximates w(W ).




















and observe that for large N, ∑Ni=−N w(xi )

2N+1 closely approximates w(W ).

There exist cycles C1, . . . ,Ct such that∑N

i=−N w(xi)

2N + 1is closely approximated by a

convex combination of w(C1), . . . ,w(Ct ).

Thus, in the limit, w(W ) ≤ max{ w(C) : C is a cycle in G}.



and observe that for large N, ∑Ni=−N w(xi )

2N+1 closely approximates w(W ).

There exist cycles C1, . . . ,Ct such that∑N

i=−N w(xi)

2N + 1is closely approximated by a

convex combination of w(C1), . . . ,w(Ct ).

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) ofstates 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 thestate diagram, and the density of X equals the average weight of itscorresponding walk, WX .

By the cycle lemma, the maximum density of an infinite walk in the state diagramis achieved by repeating a simple cycle C. The independent set XC given by thosestates has maximum density δ(XC) = α(S).

The length of C is at most 2s, so the period of XC is at most s2s.



By the cycle lemma, the maximum density of an infinite walk in the state diagramis achieved by repeating a simple cycle C.

The independent set XC given by thosestates has maximum density δ(XC) = α(S).










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) isachieved 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, Erdos, and Skilton, ’90) For k = χ(G(S)), there exists aperiodic proper k -coloring c with minimum period at most sks.

Example Theorem (S = {1,2, k})

Theorem (Zhu, ’02)

α({1,2, k}) =

k

3k+3 if k ≡ 0 (mod 3)

13 if k ≡ 1 (mod 3)

13 if k ≡ 2 (mod 3).


Theorem (CGHRS, ’14+) For k > 4,

α({1,4, k}) =

2k5k+5 if k ≡ 0 (mod 5)

25 if k ≡ 1 (mod 5)

2k+15k+5 if k ≡ 2 (mod 5)

2k−15k+5 if k ≡ 3 (mod 5)

25 if k ≡ 4 (mod 5).


0 20 40 60 80 100

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0 mod 5

1 mod 5

2 mod 5

3 mod 5

4 mod 5

α({1,4, k}) =

2k5k+5 if k ≡ 0 (mod 5)

25 if k ≡ 1 (mod 5)

2k+15k+5 if k ≡ 2 (mod 5)

2k−15k+5 if k ≡ 3 (mod 5)

25 if k ≡ 4 (mod 5).

−→ 25

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 | − bdischarge // µ∗(Bj)

defines��

Stage 2: Frames ν∗(Fj)discharge // ν′(Fj) ≥ c

If ν′(Fj) ≥ c for all frames, then

δ(X ) ≤ atbt + c

.




defines��



δ(X ) ≤ atbt + c

.




defines��



δ(X ) ≤ atbt + c

.

Example Discharging Argument (S = {1,4, k})Theorem (CGHRS, ’14+) For k > 4,

α({1,4, k}) =

2k5k+5 if k ≡ 0 (mod 5)25 if k ≡ 1 (mod 5)2k+15k+5 if k ≡ 2 (mod 5)2k−15k+5 if k ≡ 3 (mod 5)25 if k ≡ 4 (mod 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 2 3

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 2 3

Block Size µ-charge µ∗-charge

2 −1 0

3 1 0, 1

5 5 4, 5

6 7 6, 7

Example Discharging Argument (S = {1,4, k})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 2 3

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 2 3

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

Example Discharging Argument (S = {1,4, k})Case k = 5i + 3:

Let t = 2i + 1 and c = 3. Thus atbt+c = 4i+2

10i+8 = 2i+15i+4 = 2k−1

5k+5 .

We use the followingsecond-stage discharging rule.

(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.

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.



10i+8 = 2i+15i+4 = 2k−1

5k+5 . We use the followingsecond-stage discharging rule.








10i+8 = 2i+15i+4 = 2k−1









10i+8 = 2i+15i+4 = 2k−1







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 A. J. RadcliffeDerrick Stolee∗

Iowa State [email protected]

http://www.math.iastate.edu/dstolee/r/distance.htm

January 18th, 2014AMS/MAA Joint Math Meetings

Documents

James M. Carraher David Galvin Stephen G. Hartke A. J. Radcliffe …orion.math.iastate.edu/dstolee/presentations/2014-01-JMM.pdf · 2014-01-18 · James M. Carraher David Galvin University