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

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

Anthony BonatoRyerson University

Toronto, Canada

6th Combinatorics Day @ LethbridgeMarch 28, 2009

04/21/23 Anthony Bonato 2

• window graph, grid graph, Paley graph P9, K3 □ K3, …

• vertex-transitive, edge-transitive, self-complementary, SRG(9,4,1,2)


• 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

• 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


• affine plane of order 3

• colours represent parallel classes

• point graph when we remove two parallel classes: P9


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


1-e.c. graphs

• no universal nor isolated vertices

• egs:– paths– cycles– matchings, …


2-e.c. graphs


A 3-e.c. graph


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


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)


2. Models for the web graphand complex networks


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.


Existence

• not obvious from the definition that n-e.c. graphs exist for all n

• an elementary proof of this uses random graphs


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

51 2 3 4


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.


Proof for p = 1/2

• the probability that G(m,1/2) is not n-e.c. is bounded above by

).1(2

112 o

n

m nm

nn


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


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)


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


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 mec(n) to be the smallest order of an n-

e.c. graph– compute exact values of mec, and study

asymptotics


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)


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)


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


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


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))


Example1 2 3

4 5 6

7 8 9


Example


Random slopes

x

y

z

• toss a coin (blue = heads, red = tails) to determine which slopes to include in U

LS


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


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.


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


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

– projection πq(x) is the slope of the block containing x,q

– for a set X, πq(X) defined analogously


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|


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


Proof continued

• By Stirling’s formula we obtain that


2. Orders

• mec(n) = minimum order of an n-e.c. graph

• mec(1) = 4

• mec(2) = 9

• no other values known!


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 boundmec(n) = O(n22n)


Open problem

• what is the asymptotic order of mec(n) ?

• (Caccetta, Erdős, Vijayan, 85) conjectured that the following limit exists:

nec

n n

nm

2

)(lim


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



N-e.c. tournaments

• n-e.c. tournaments

• a 2-e.c. tournament T7 :

A B

z


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)


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}


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).


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

order #

8 0

9 14

10 1083

order #

19 1

20 0

21 0

22 0

2-e.c. 3-e.c.


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


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


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.


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


Sketch of proof


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 Nc(x) are e.c. so isomorphic to R

• Conjecture: R is the only countable graph with (♥).


• preprints, reprints, contact:

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

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