1 / 54

The Search for N-e.c. Graphs and Tournaments

6 th Combinatorics Day @ Lethbridge March 28, 2009. The Search for N-e.c. Graphs and Tournaments. Anthony Bonato Ryerson University Toronto, Canada. window graph , grid graph , Paley graph P 9 , K 3 □ K 3 , … vertex-transitive, edge-transitive, self-complementary, SRG(9,4,1,2).

laurie
Télécharger la présentation

The Search for N-e.c. Graphs and Tournaments

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 6th Combinatorics Day @ Lethbridge March 28, 2009 The Search for N-e.c. Graphs and Tournaments Anthony Bonato Ryerson University Toronto, Canada

  2. window graph, grid graph, Paley graph P9, K3 □ K3, … • vertex-transitive, edge-transitive, self-complementary, SRG(9,4,1,2) Anthony Bonato

  3. 2-e.c. adjacency property: for each pair of vertices x,y, there are vertices joined to x,y in all the four possible ways Anthony Bonato

  4. 2-e.c. adjacency property: for each pair of vertices x,y, there are vertices joined to x,y in all the four possible ways • unique minimum order 2-e.c. graph Anthony Bonato

  5. affine plane of order 3 • colours represent parallel classes • point graph when we remove two parallel classes: P9 Anthony Bonato

  6. n-existentially closed graphs • fix n a positive integer • a graph G is n-existentially closed (n-e.c.) if for each n-set X in V(G) and every partition of X into A, B, there is a vertex not in X joined to each vertex of A, and to no vertex of B B A z Anthony Bonato

  7. 1-e.c. graphs • no universal nor isolated vertices • egs: • paths • cycles • matchings, … Anthony Bonato

  8. 2-e.c. graphs Anthony Bonato

  9. A 3-e.c. graph Anthony Bonato

  10. Connections with logic • existential closure was introduced by Abraham Robinson in 1960’s • gives a generalization of algebraically closed fields to first-order structures • (Fagin,76) used adjacency properties analogous to n-e.c. to prove the 0-1 law for the first-order theory of graphs Anthony Bonato

  11. Recent applications of n-e.c. graphs:1. Cop number of random graphs R C (B, Hahn, Wang, 07) c(G(m,p)) = Θ(log m) Anthony Bonato

  12. 2. Models for the web graphand complex networks Anthony Bonato

  13. Properties of n-e.c. graphs • suppose that G is n-e.c. with n>1 • the complement of G is n-e.c. • |V(G)| = Ω(2n), |E(G)| = Ω(n2n) • for all vertices x, the subgraph induced by N(x) and Nc(x) are (n-1)-e.c. Anthony Bonato

  14. Existence • not obvious from the definition that n-e.c. graphs exist for all n • an elementary proof of this uses random graphs Anthony Bonato

  15. G(m,p)(Erdős, Rényi, 63) • m a positive integer, p = p(m) a real number in (0,1) • G(m,p): probability space on graphs with nodes {1,…,m}, two nodes joined independently and with probability p 4 1 2 3 5 Anthony Bonato

  16. Random graphs are n-e.c. • an event A holds in G(m,p)asymptotically almost surely (a.a.s.) if A holds with probability tending to 1 as m → ∞ Theorem (Erdős, Rényi, 63) For n > 0 fixed, a.a.s. G(m,p) is n-e.c. Anthony Bonato

  17. Proof for p = 1/2 • the probability that G(m,1/2) is not n-e.c. is bounded above by Anthony Bonato

  18. Determinism • few examples of explicit n-e.c. graphs are known • difficulty arises for large n (even n > 2 a problem) • one family that has all the n-e.c. properties are Paley graphs Anthony Bonato

  19. Paley graphs Pq • fix q prime power congruent to 1(mod 4) • vertices: GF(q) • edges: x and y are joined iff x-y is a non-zero quadratic residue (square) Anthony Bonato

  20. Paley graphs are n-e.c. • properties of Paley graphs: • self-complementary, symmetric • SRG(q,(q-1)/2,(q-5)/4,(q-1)/4) • (Bollobás, Thomason, 81) If q > n222n-2, then Pq is n-e.c. • proof relies on Riemann hypothesis for finite fields Anthony Bonato

  21. Research directions • Constructions • construct explicit examples of n-e.c. graphs • difficult even for n = 3 • proofs usually rely on techniques from other disciplines: algebra, number theory, matrix theory, logic, design theory, … • Orders • define mec(n) to be the smallest order of an n-e.c. graph • compute exact values of mec, and study asymptotics Anthony Bonato

  22. 1. Constructions • Paley graphs (Bollobás, Thomason, 81), (Blass, Exoo, Harary, 81), and their variants (eg cubic Paley graphs) • 2-e.c. vertex- and edge-critical graphs (B, K.Cameron,01) • 3-e.c. SRG from Bush-type Hadamard matrices (B,Holzman,Kharaghani,01) • exponentially many n-e.c. SRG (Cameron, Stark, 02) • n-e.c. graphs from matrices and constraints (Blass, Rossman, 05) • 2-e.c. graphs from block intersection graphs • Steiner triple systems: (Forbes, Grannell, Griggs, 05) • balanced incomplete block designs (McKay, Pike, 07) • n-e.c. graphs from affine planes (Baker, B, Brown, Szőnyi, 08) Anthony Bonato

  23. New construction: Steiner 2-designs • Steiner 2-design,S(2,k,v): k-subsets or blocks of a v-set of points, so that each distinct pair of points is contained in a unique block • a 2-(v,k,1) design • examples: • Steiner triple systems 2-(v,3,1) • affine planes 2-(q2,q,1) • affine spaces 2-(qm,q,1) Anthony Bonato

  24. Resolvability • a Steiner 2-design is resolvable if its blocks may be partitioned into parallel classes, so each point is in a unique block of each parallel class • examples of resolvable Steiner 2-designs: Kirkman triple systems Anthony Bonato

  25. Line at infinity • for each affine plane of order q, the line at infinity has order q+1, and corresponds to slopes of lines • generalizes to resolvable Steiner 2-designs • label each parallel class; labels called slopes • set of (v-1)/(k-1) labels is the denoted by LS Anthony Bonato

  26. Slope graphs • for a set U of slopes in a S(v,k,2), define G(U) so that vertices are points, and two vertices x and y are adjacent if the slope of the line xy is in U • graphs G(U): slope graphs • G(U) is regular with degree |U|(k-1) • introduced for affine planes by Delsarte, Goethals, and Turyn • for affine planes: SRG(q2,|U|(k-1),k-2+(|U|-1)(|U|-2),|U|(|U|-1)) Anthony Bonato

  27. Example 3 2 1 4 5 6 9 8 7 Anthony Bonato

  28. Example Anthony Bonato

  29. Random slopes • toss a coin (blue = heads, red = tails) to determine which slopes to include in U LS y z x Anthony Bonato

  30. The space G(v,S,p) • given S = S(v,k,2) choose m from LS to be in U independently with probability p (where p = p(v) can be a function of v) • obtain a probability space G(v,S,p) • obtain regular graphs • Chernoff bounds: G(v,S,p) is regular with degree concentrated on pv Anthony Bonato

  31. New result Theorem (Baker, B, McKay, Prałat, 09) Let S = S(v,k,2) be an acceptable Steiner 2-design (i.e. k = O(v2)). Then a.a.s. G(v,S,p) is n-e.c. for all n = n(v) = 1/2log1/pv - 5log1/plogv. Anthony Bonato

  32. Discussion • construction gives sparse n-e.c. graphs: if p = v-1/loglogv then the degrees concentrate on v1-1/loglogv = o(v) and n = (1+o(1))1/2loglogv Anthony Bonato

  33. Sketch of proof • fix X a set of n points • estimate probability there is no q correctly joined to X • problem: given two distinct q1 and q2, probability q1 and q2 correctly joined to X is NOT independent • the proof relies on the template lemma • gives a pool of points PX with desirable independence properties • projectionπq(x) is the slope of the block containing x,q • for a set X, πq(X) defined analogously Anthony Bonato

  34. Template Lemma • items (1,2): for any two points q1 and q2 in PX, projections are distinct n-sets; gives independence • item (3):PX is large enough with s= |PX| Anthony Bonato

  35. Proof continued • given a partition of X into A,B with |B|=b, the probability pn that there is no vertex q in PX correctly joined to X is Anthony Bonato

  36. Proof continued • By Stirling’s formula we obtain that Anthony Bonato

  37. 2. Orders • mec(n)= minimum order of an n-e.c. graph • mec(1)= 4 • mec(2)= 9 • no other values known! Anthony Bonato

  38. Bounds • directly: mec(3) ≥ 20 • computer search: mec(3)≤ 28 • mec(3) ≥ 24 (Gordinowicz, Prałat, 09) • 15,000 hours on 8000+ CPUs (!) • (Caccetta, Erdős, Vijayan, 85): mec(n) = Ω(n2n) • random graphs give best known upper bound mec(n) = O(n22n) Anthony Bonato

  39. Open problem • what is the asymptotic order of mec(n) ? • (Caccetta, Erdős, Vijayan, 85) conjectured that the following limit exists: Anthony Bonato

  40. Possible orders • for which m do m-vertexn-e.c. graphs exist? • (Caccetta, Erdős, Vijayan, 85): 2-e.c. graphs exist for all orders m ≥ 9 • (Gordinowicz, Prałat, 09), (Pikhurko,Singh,09): a 3-e.c. graph of order n might not exist onlyif n = 24, 25, 26, 27, 30, 31, 33 Anthony Bonato

  41. Tournaments Anthony Bonato

  42. N-e.c. tournaments • n-e.c. tournaments • a 2-e.c. tournament T7 : B A z Anthony Bonato

  43. Explicit constructions • existence: probabilistic method • explicit constructions: • Paley tournaments Tq(Graham, Spencer, 71) • q congruent to 3 (mod 4) • 2-e.c. vertex- and edge-critical tournaments (B, K.Cameron, 06) • n-e.c. tournaments from matrices and constraints (Blass, Rossman, 05) Anthony Bonato

  44. New construction: circulant tournaments • fix m > 0, and work (mod 2m+1) • choose J in {1,…,2m} such that j in J iff –j is not in J • circulant tournamentT(J) has vertices the residues (mod 2m+1) and directed edges (i,j) if i – j is in J J = {1,2,4} Anthony Bonato

  45. Random circulants • T(J) is vertex-transitive (and so regular) • randomize the selection of J: for p fixed, add j in {1,…,m}; with probability 1-p add -j • obtain probability space CT(m,p) Theorem (B,Gordinowicz,Prałat,09) A.a.s. CT(m,p) is n-e.c. with n = log1/pm - 4log1/plogm-O(1). Anthony Bonato

  46. Minimum orders • tec(n)= minimum order of an n-e.c. tournament • (B,K.Cameron,06): tec(1)= 3, tec(2)= 7 (directed cycle, T7, respectively) • (B,Gordinowicz,Prałat,09):tec(3)= 19 2-e.c. 3-e.c. Anthony Bonato

  47. Bounds • (Szekeres, Szekeres,65) and random tournaments give: Ω(n2n) = tec(n) = O(n22n) • order of tec(n) is unknown • (BGP,09): 47 ≤ tec(4) ≤ 67 111 ≤ tec(5) ≤ 359 Anthony Bonato

  48. The infinite random graph Theorem (Erdős,Rényi, 63): With probability 1, any two graphs sampled from G(N,p) are isomorphic. • isotype R unique countable graph with the e.c. property: n-e.c. for all n> 0 Anthony Bonato

  49. An geometric representation of R • define a graph G(p) with vertices the points with rational coordinates in the plane, edges determined by lines with randomly chosen slopes • with probability 1, G(p) is e.c. Anthony Bonato

  50. Explicit slope sets • (BBMP,09): slope sets that are the union of finitely many intervals are 3-e.c., but not 4-e.c. • problem: find explicit slope sets that give rise to an n-e.c. graph for each n ≥ 4 Anthony Bonato

More Related