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

8/28/2015Anthony Bonato2 window graph, grid graph, Paley graph P 9, K 3 □ K 3, … vertex-transitive, edge-transitive, self-complementary, SRG(9,4,1,2)

8/28/2015Anthony Bonato3 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

8/28/2015Anthony Bonato4 unique minimum order 2-e.c. graph 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

8/28/2015Anthony Bonato5 affine plane of order 3 colours represent parallel classes point graph when we remove two parallel classes: P 9

8/28/2015Anthony Bonato6 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 A B z

8/28/2015Anthony Bonato7 1-e.c. graphs no universal nor isolated vertices egs: – paths – cycles – matchings, …

8/28/2015Anthony Bonato8 2-e.c. graphs

8/28/2015Anthony Bonato9 A 3-e.c. graph

8/28/2015Anthony Bonato10 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

8/28/2015Anthony Bonato11 Recent applications of n-e.c. graphs: 1. Cop number of random graphs C R (B, Hahn, Wang, 07) c(G(m,p)) = Θ(log m)

8/28/2015Anthony Bonato12 2. Models for the web graph and complex networks

8/28/2015Anthony Bonato13 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(2 n ), |E(G)| = Ω(n2 n ) –for all vertices x, the subgraph induced by N(x) and N c (x) are (n-1)-e.c.

8/28/2015Anthony Bonato14 Existence not obvious from the definition that n-e.c. graphs exist for all n an elementary proof of this uses random graphs

8/28/2015Anthony Bonato15 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

8/28/2015Anthony Bonato16 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.

8/28/2015Anthony Bonato17 Proof for p = 1/2 the probability that G(m,1/2) is not n-e.c. is bounded above by

8/28/2015Anthony Bonato18 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

8/28/2015Anthony Bonato19 Paley graphs P q 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)

8/28/2015Anthony Bonato20 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 > n 2 2 2n-2, then P q is n-e.c. –proof relies on Riemann hypothesis for finite fields

8/28/2015Anthony Bonato21 Research directions 1.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, … 2.Orders –define m ec (n) to be the smallest order of an n- e.c. graph –compute exact values of m ec, and study asymptotics

8/28/2015Anthony Bonato22 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)

8/28/2015Anthony Bonato23 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-(q 2,q,1) –affine spaces 2-(q m,q,1)

8/28/2015Anthony Bonato24 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

8/28/2015Anthony Bonato25 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 L S

8/28/2015Anthony Bonato26 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(q 2,|U|(k-1),k-2+(|U|-1)(|U|-2),|U|(|U|-1))

8/28/2015Anthony Bonato27 Example

8/28/2015Anthony Bonato28 Example

8/28/2015Anthony Bonato29 Random slopes x y z toss a coin (blue = heads, red = tails) to determine which slopes to include in U LSLS

8/28/2015Anthony Bonato30 The space G(v,S,p) given S = S(v,k,2) choose m from L S 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

8/28/2015Anthony Bonato31 New result Theorem (Baker, B, McKay, Prałat, 09) Let S = S(v,k,2) be an acceptable Steiner 2- design (i.e. k = O(v 2 )). Then a.a.s. G(v,S,p) is n-e.c. for all n = n(v) = 1/2log 1/p v - 5log 1/p logv.

8/28/2015Anthony Bonato32 Discussion construction gives sparse n-e.c. graphs: if p = v -1/loglogv then the degrees concentrate on v 1-1/loglogv = o(v) and n = (1+o(1))1/2loglogv

8/28/2015Anthony Bonato33 Sketch of proof fix X a set of n points estimate probability there is no q correctly joined to X problem: given two distinct q 1 and q 2, probability q 1 and q 2 correctly joined to X is NOT independent the proof relies on the template lemma –gives a pool of points P X with desirable independence properties –projection π q (x) is the slope of the block containing x,q –for a set X, π q (X) defined analogously

8/28/2015Anthony Bonato34 Template Lemma items (1,2): for any two points q 1 and q 2 in P X, projections are distinct n-sets; gives independence item (3): P X is large enough with s= |P X |

8/28/2015Anthony Bonato35 Proof continued given a partition of X into A,B with |B|=b, the probability p n that there is no vertex q in P X correctly joined to X is

8/28/2015Anthony Bonato36 Proof continued By Stirling’s formula we obtain that

8/28/2015Anthony Bonato37 2. Orders m ec (n) = minimum order of an n-e.c. graph m ec (1) = 4 m ec (2) = 9 no other values known!

8/28/2015Anthony Bonato38 Bounds directly: m ec (3) ≥ 20 computer search: m ec (3) ≤ 28 m ec (3) ≥ 24 (Gordinowicz, Prałat, 09) –15,000 hours on CPUs (!) (Caccetta, Erdős, Vijayan, 85): m ec (n) = Ω(n2 n ) random graphs give best known upper bound m ec (n) = O(n 2 2 n )

8/28/2015Anthony Bonato39 Open problem what is the asymptotic order of m ec (n) ? (Caccetta, Erdős, Vijayan, 85) conjectured that the following limit exists:

8/28/2015Anthony Bonato40 Possible orders for which m do m-vertex n-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 only if n = 24, 25, 26, 27, 30, 31, 33

Tournaments 8/28/2015Anthony Bonato41

8/28/2015Anthony Bonato42 N-e.c. tournaments n-e.c. tournaments a 2-e.c. tournament T 7 : A B z

8/28/2015Anthony Bonato43 Explicit constructions existence: probabilistic method explicit constructions: –Paley tournaments T q (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)

8/28/2015Anthony Bonato44 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 tournament T(J) has vertices the residues (mod 2m+1) and directed edges (i,j) if i – j is in J J = {1,2,4}

8/28/2015Anthony Bonato45 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 = log 1/p m - 4log 1/p logm-O(1).

8/28/2015Anthony Bonato46 Minimum orders t ec (n) = minimum order of an n-e.c. tournament (B,K.Cameron,06): t ec (1) = 3, t ec (2) = 7 (directed cycle, T 7, respectively) (B,Gordinowicz,Prałat,09): t ec (3) = 19 order# order# e.c. 3-e.c.

8/28/2015Anthony Bonato47 Bounds (Szekeres, Szekeres,65) and random tournaments give: Ω(n2 n ) = t ec (n) = O(n 2 2 n ) order of t ec (n) is unknown (BGP,09): 47 ≤ t ec (4) ≤ ≤ t ec (5) ≤ 359

8/28/2015Anthony Bonato48 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 The infinite random graph

8/28/2015Anthony Bonato49 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.

8/28/2015Anthony Bonato50 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

8/28/2015Anthony Bonato51 Sketch of proof

8/28/2015Anthony Bonato52 Neighbours in R the unique countable e.c. graph R has a peculiar robustness property: (♥): for each vertex x, the subgraph induced by N(x) and N c (x) are e.c. so isomorphic to R Conjecture: R is the only countable graph with (♥).

8/28/2015Anthony Bonato53

8/28/2015Anthony Bonato54 preprints, reprints, contact: Google: “Anthony Bonato”