Slides - Iowa State University
Transcription
Slides - Iowa State University
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 University of Nebraska–Lincoln [email protected] A. J. Radcliffe University of Nebraska–Lincoln [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 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 The independence ratio α(S ) is the supremum of δ(A) over all independent sets A in G (S ). 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 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 . 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 = d1 . 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 = d1 . Every vertex appears in d sets Xi , so d · 1 d ≥ 1. 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 = 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 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 wi wi +1 is an edge for all i ∈ Z. 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 wi wi +1 is an edge for all i ∈ Z. The (upper) average weight of W is w (W ) = lim supN →∞ ∑N i =−N wi 2N +1 . 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 wi wi +1 is an edge for 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 ). Proof of Cycle Lemma 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 Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Proof of Cycle Lemma Let W = (xi )i ∈Z be an infinite walk in G and let N n(G ). Let WN = (xi )N i =−N and observe that for large N, ∑N i =−N w (xi ) 2N +1 closely approximates w (W ). ∑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 Proof of Cycle Lemma Let W = (xi )i ∈Z be an infinite walk in G and let N n(G ). Let WN = (xi )N i =−N and observe that for large N, ∑N i =−N w (xi ) 2N +1 closely approximates w (W ). ∑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 Proof of Periodic Sets Proof of Periodic Sets Proof of Periodic Sets Proof of Periodic Sets Proof of Periodic Sets Proof of Periodic Sets Proof of Periodic Sets A state σ is a subset of {0, . . . , s − 1} (there are 2s such states). Proof of Periodic Sets A state σ is a subset of {0, . . . , s − 1} (there are 2s such states). Proof of Periodic Sets A state is allowed if σ is independent in G (S ). Proof of Periodic Sets Proof of Periodic Sets 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 ). Proof of Periodic Sets 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 . Proof of Periodic Sets 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. Proof of Periodic Sets 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. The independent set XC given by those states has maximum density δ(XC ) = α(S ). Proof of Periodic Sets 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. 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). Example Theorem (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 ≡ 0 (mod 5) if k ≡ 1 (mod 5) if k ≡ 2 (mod 5) if k ≡ 3 (mod 5) if k ≡ 4 (mod 5). Example Theorem (S = {1, 4, k }) 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 ). 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 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 ) 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 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 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 Example Discharging Argument (S = {1, 4, k }) 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 . Example Discharging Argument (S = {1, 4, k }) Case k = 5i + 3: Let t = 2i + 1 and c = 3. Thus 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 . Example Discharging Argument (S = {1, 4, k }) Case k = 5i + 3: Let t = 2i + 1 and c = 3. Thus 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 . It remains to show that: 1. if ν∗ (Fj ) < 3 = c, then σ(Fj ) = 5i + 2 and Fj pulls charge by (S2). Example Discharging Argument (S = {1, 4, k }) Case k = 5i + 3: Let t = 2i + 1 and c = 3. Thus 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 . 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 } Densities for S = {1, 1 + k , 1 + k + i } 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
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