james m. carraher david galvin stephen g. hartke a. j. radcliffe...
TRANSCRIPT
![Page 1: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/2.jpg)
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]
![Page 3: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/3.jpg)
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.
![Page 4: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/4.jpg)
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.
![Page 5: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/5.jpg)
0-1 1
-2 2
-3 3
-4 4
-5 5-6
G(12, {1,4,5})
![Page 6: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/6.jpg)
0-1 1
-2 2
-3 3
-4 4
-5 5
-6 6
G(18, {1,4,5})
![Page 7: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/7.jpg)
0-1 1
-2 2
-3 3
-4 4
-5 5
-6 6
G(28, {1,4,5})
![Page 8: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/8.jpg)
0-1 1
-2 2
-3 3
-4 4
-5 5
-6 6
G(40, {1,4,5})
![Page 9: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/9.jpg)
0-1 1
-2 2
-3 3
-4 4
-5 5
G(60, {1,4,5})
![Page 10: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/10.jpg)
0-1 1-2 2
-3 3
-4 4
-5 5
G(128, {1,4,5})
![Page 11: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/11.jpg)
0-1 1-2 2
-3 3
-4 4
G(256, {1,4,5})
![Page 12: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/12.jpg)
0-1 1-2 2-3 3
-4 4
G(512, {1,4,5})
![Page 13: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/13.jpg)
0-1 1-2 2-3 3-4 4
G(1024, {1,4,5})
![Page 14: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/14.jpg)
0-1 1-2 2-3 3-4 4
G(2048, {1,4,5})
![Page 15: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/15.jpg)
-4 -3 -2 -1 0 1 2 3 4
G(∞, {1,4,5}) = G({1,4,5})
![Page 16: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/16.jpg)
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.
![Page 17: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/17.jpg)
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.
![Page 18: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/18.jpg)
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.
![Page 19: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/19.jpg)
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.
![Page 20: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/20.jpg)
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).
![Page 21: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/21.jpg)
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).
![Page 22: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/22.jpg)
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).
![Page 23: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/23.jpg)
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 )
.
![Page 24: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/24.jpg)
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 · 1d ≥ 1.
Then
χf (G(S)) ≤ ∑p−1i=0 cXi =
pd =
1δ(X )
.
![Page 25: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/25.jpg)
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 )
.
![Page 26: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/26.jpg)
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 )
.
![Page 27: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/27.jpg)
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).
![Page 28: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/28.jpg)
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.
![Page 29: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/29.jpg)
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.
![Page 30: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/30.jpg)
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.
![Page 31: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/31.jpg)
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.
![Page 32: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/32.jpg)
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 ).
![Page 33: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/33.jpg)
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 ).
![Page 34: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/34.jpg)
Proof of Cycle Lemma
![Page 35: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/35.jpg)
Proof of Cycle Lemma
![Page 36: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/36.jpg)
Proof of Cycle Lemma
![Page 37: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/37.jpg)
Proof of Cycle Lemma
![Page 38: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/38.jpg)
Proof of Cycle Lemma
![Page 39: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/39.jpg)
Proof of Cycle Lemma
![Page 40: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/40.jpg)
Proof of Cycle Lemma
![Page 41: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/41.jpg)
Proof of Cycle Lemma
![Page 42: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/42.jpg)
Proof of Cycle Lemma
![Page 43: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/43.jpg)
Proof of Cycle Lemma
![Page 44: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/44.jpg)
Proof of Cycle Lemma
![Page 45: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/45.jpg)
Proof of Cycle Lemma
![Page 46: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/46.jpg)
Proof of Cycle Lemma
![Page 47: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/47.jpg)
Proof of Cycle Lemma
![Page 48: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/48.jpg)
Proof of Cycle Lemma
![Page 49: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/49.jpg)
Proof of Cycle Lemma
![Page 50: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/50.jpg)
Proof of Cycle Lemma
![Page 51: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/51.jpg)
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, ∑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}.
![Page 52: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/52.jpg)
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, ∑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}.
![Page 53: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/53.jpg)
Proof of Periodic Sets
![Page 54: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/54.jpg)
Proof of Periodic Sets
![Page 55: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/55.jpg)
Proof of Periodic Sets
![Page 56: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/56.jpg)
Proof of Periodic Sets
![Page 57: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/57.jpg)
Proof of Periodic Sets
![Page 58: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/58.jpg)
Proof of Periodic Sets
![Page 59: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/59.jpg)
Proof of Periodic Sets
![Page 60: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/60.jpg)
Proof of Periodic Sets
A state σ is a subset of {0, . . . , s− 1} (there are 2s such states).
![Page 61: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/61.jpg)
Proof of Periodic Sets
A state σ is a subset of {0, . . . , s− 1} (there are 2s such states).
![Page 62: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/62.jpg)
Proof of Periodic Sets
A state is allowed if σ is independent in G(S).
![Page 63: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/63.jpg)
Proof of Periodic Sets
![Page 64: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/64.jpg)
Proof of Periodic Sets
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).
![Page 65: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/65.jpg)
Proof of Periodic Sets
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.
![Page 66: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/66.jpg)
Proof of Periodic Sets
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.
![Page 67: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/67.jpg)
Proof of Periodic Sets
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.
![Page 68: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/68.jpg)
Proof of Periodic Sets
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.
![Page 69: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/69.jpg)
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.
![Page 70: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/70.jpg)
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).
![Page 71: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/71.jpg)
Example Theorem (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).
![Page 72: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/72.jpg)
Example Theorem (S = {1,4, k})
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
![Page 73: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/73.jpg)
Discharging
Suppose X = {xi : i ∈ Z} ⊆ Z is an infinite independent set in G(S).
![Page 74: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/74.jpg)
Discharging
Elements are labeled . . . , x−1, x0, x1, . . . , xi , . . . .
![Page 75: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/75.jpg)
Discharging
Blocks are sets Bk = {xk , xk + 1, . . . , xk+1 − 1}.(“Intervals” closed on element xk and open on xk+1)
![Page 76: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/76.jpg)
Discharging
Frames are collections Fj = {Bj ,Bj+1, . . . ,Bj+t−1}.(There are t blocks in each frame.)
![Page 77: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/77.jpg)
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
.
![Page 78: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/78.jpg)
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
.
![Page 79: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/79.jpg)
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
.
![Page 80: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/80.jpg)
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
![Page 81: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/81.jpg)
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
![Page 82: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/82.jpg)
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.
![Page 83: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/83.jpg)
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.
![Page 84: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/84.jpg)
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.
![Page 85: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/85.jpg)
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.
![Page 86: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/86.jpg)
Densities for S = {1,1 + k ,1 + k + i}
![Page 87: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/87.jpg)
Densities for S = {1,1 + k ,1 + k + i}
![Page 88: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/88.jpg)
Densities for S = {1,1 + k ,1 + k + i}
![Page 89: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/89.jpg)
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?
![Page 90: 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](https://reader034.vdocument.in/reader034/viewer/2022042123/5e9db510c1b4c31b7c50e25e/html5/thumbnails/90.jpg)
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