Modelling and Searching Networks Lecture 9 – Meyniel’s Conjecture Miniconference on the Mathematics of Computation MTH 707 Modelling and Searching Networks Lecture 9 – Meyniel’s Conjecture Dr. Anthony Bonato Ryerson University

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).

Henri Meyniel, courtesy Geňa Hahn

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).

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

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.

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

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).

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:

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!

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

Example Fano plane Heawood graph

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

IG of projective planes are ME Lemma 10.2. 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.

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.