the search for n-e.c. graphs and tournaments anthony bonato ryerson university toronto, canada 6 th...
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The Search for N-e.c. Graphs and Tournaments
Anthony BonatoRyerson University
Toronto, Canada
6th Combinatorics Day @ LethbridgeMarch 28, 2009
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• window graph, grid graph, Paley graph P9, K3 □ K3, …
• vertex-transitive, edge-transitive, self-complementary, SRG(9,4,1,2)
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• 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
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• 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
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• affine plane of order 3
• colours represent parallel classes
• point graph when we remove two parallel classes: P9
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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
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1-e.c. graphs
• no universal nor isolated vertices
• egs:– paths– cycles– matchings, …
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2-e.c. graphs
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A 3-e.c. graph
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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
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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)
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2. Models for the web graphand complex networks
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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.
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Existence
• not obvious from the definition that n-e.c. graphs exist for all n
• an elementary proof of this uses random graphs
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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
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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.
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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
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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
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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)
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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
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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
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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)
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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)
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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
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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
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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))
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Example1 2 3
4 5 6
7 8 9
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Example
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Random slopes
x
y
z
• toss a coin (blue = heads, red = tails) to determine which slopes to include in U
LS
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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
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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.
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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
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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
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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|
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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
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Proof continued
• By Stirling’s formula we obtain that
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2. Orders
• mec(n) = minimum order of an n-e.c. graph
• mec(1) = 4
• mec(2) = 9
• no other values known!
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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)
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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
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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
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N-e.c. tournaments
• n-e.c. tournaments
• a 2-e.c. tournament T7 :
A B
z
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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)
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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}
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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).
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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.
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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
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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
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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.
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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
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Sketch of proof
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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 (♥).
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• preprints, reprints, contact:
Google: “Anthony Bonato”