1. Week 10 – Meyniel’s conjecture Dr. Anthony Bonato Ryerson University AM8002 Fall 2014

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(n 1/2 ). 2

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4. 4 Henri Meyniel, courtesy Geňa Hahn

5. 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). 5

6. State-of-the-art (Lu, Peng, 12) proved that –independently proved by (Scott, Sudakov,11) and (Frieze, Krivelevich, Loh, 11) (Prałat,Wormald,14+): proved Meyniel’s conjecture for random graphs G(n,p) for wide range of p 6

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. 7

8. Graph classes, continued fix a graph H. A graph is H-free if it does not contain H as a subgraph example: H = K 3 ; triangle-free graphs (Joret et al,10): H-free class graphs have bounded cop number iff each component of H is a tree with at most 3 leaves. 8

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). 9

10. 10 Projective planes 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 q 2 +q+1 points; 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! 11

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 12

13. Example 13 Fano plane Heawood graph

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

15. IG of projective planes are ME Lemma 10.2. If P is a projective plane, then G(P) has girth 6, and is (q+1)-regular. Corollary 10.3: IG of projective planes are ME. 15

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

17. Diameter 2 (Lu, Peng, 12): If G has diameter 2, then c(G) ≤ 2n 1/2 - 1. –diameter 2 graphs satisfy Meyniel’s conjecture proof uses the probabilistic method Question: are there explicit Meyniel extremal families whose members are diameter two? 17

18. Polarity graphs suppose PG(2,q) has points P and lines L. A polarity is a function π: P→ L such that for all points p,q, p π(q) iff q π(p). eg of orthogonal polarity: point mapped to its orthogonal complement polarity graph: vertices are points, x and y adjacent if x π(y) 18

19. Properties of polarity graphs order q 2 +q+1 (q,q+1)-regular C 4 -free diameter 2 19

20. Meyniel Extremal Theorem 10.4 (Bonato,Burgess,13) Let q be a prime power. If G q is a polarity graph of PG(2, q), then q/2 ≤ c(G q ) ≤ q + 1. lower bound: lemma 20

21. Lower bounds Lemma 10.5 (Aigner,Fromme, 84) If G is a connected graph of girth at least 5, then c(G) ≥ δ(G). Lemma 10.6 (Bonato, Burgess,13) If G is connected and K 2,t -free, then c(G) ≥ δ(G) / t. –applies to polarity graphs: t = 2 21