1. Modelling and Searching Networks Lecture 9 – Meyniel’s Conjecture
Miniconference on the Mathematics of Computation
AM 8002
Modelling and Searching Networks Lecture 9 – Meyniel’s Conjecture
Dr. Anthony Bonato
Ryerson University

2. How big can the cop number be?
if G is disconnected of order n, then we can have c(G) = n (example?)
c(n) = maximum cop number of a connected
graph of order n
Meyniel’s Conjecture: c(n) = O(n1/2).

4. Henri Meyniel, courtesy Geňa Hahn

5. c(n) = O(n loglog n / log n)
Background
(Frankl,87) stated conjecture; but only implicitly
Frankl proved
c(n) = O(n loglog n / log n)
25 years past, and the conjecture was largely forgotten
(Chinifooroshan,08) improved the bound to
c(n) = O(n / log n).

6. State-of-the-art (Lu, Peng, 12) proved that
independently proved by (Scott, Sudakov,11) and
(Frieze, Krivelevich, Loh, 11)

7. Some graph classes with small cop number
a graph is planar if it can be drawn in the plane without edge crossings
(Aigner, Fromme,84): planar graphs have cop number at most 3.

8. Exercise
10.1 Show the following planar graph has cop number 3.

9. Cop number of G(n,p) in G(n,p), the cop number of is a random variable
Theorem 10.1 (Bonato, Hahn, Wang, 07) Fix 0 < p < 1 a constant. Then a.a.s.
c(G(n,p)) = Θ(log n).

10. Projective planes eg: q=2,3: consider a finite projective plane P
two lines meet in a unique point
two points determine a unique line
exist 4 points, no line contains more than two of them
q2+q+1 points (lines); each line (point) contains (is incident with) q+1 points (lines)
eg: q=2,3:

11. Existence of projective planes
it can be proved that projective planes exist for prime power orders; it is conjectured they can only exist for these orders.
order 10 was eliminated by a heavy computer search
order 12 is open!

12. Incidence graphs let P be a projective plane
incidence graph (IG) of P:
bipartite graph G(P) with red nodes the points of P and blue nodes the lines of P
a point is joined to a line if it is on that line

13. Example
Fano plane
Heawood graph

14. Meyniel extremal families
a family of connected graphs (Gn: n ≥ 1) is Meyniel extremal (ME) if there is a constant d > 0, such that for all n ≥ 1, c(Gn) ≥ dn1/2
in other words: ME families have members with the asymptotically largest conjectured cop number

15. IG of projective planes are ME
Lemma If P is a projective plane, then G(P) has girth at least 6, and is (q+1)-regular.
Corollary 10.3: IG of projective planes are ME.

16. Exercise
10.4 Prove that for a projective plane P of order q, the incidence graph G(P) has cop number at most q+1.