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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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