1. Small subgraphs in the Achlioptas process Reto Spöhel, ETH Zürich Joint work with Torsten Mütze and Henning Thomas TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A

2. Introduction Goal: Avoid creating a copy of some fixed graph F Achlioptas process: start with the empty graph on n vertices in each step r edges are chosen uniformly at random (among all edges never seen before) select one of the r edges that is inserted into the graph, the remaining r – 1 edges are discarded How long can we avoid F by this freedom of choice? F =, r = 2

3. Introduction N 0 = N 0 ( F, r, n ) is a threshold: N 0 = N 0 ( F, r, n ) N = N ( n ) = # steps There is a strategy that avoids creating a copy of F with probability 1-o(1) N /N0N /N0 If F is a cycle, a clique or a complete bipartite graph with parts of equal size, an explicit threshold function is known. (Krivelevich, Loh, Sudakov, 2007+) Every strategy will be forced to create a copy of F with probability 1-o(1) N [N0N [N0 n 1.2 F =, r = 2 n 1.286 … r=3r=3 n 1.333 … r=4r=4 n1n1 r=1r=1

4. Our Result Theorem: Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is where We solve the problem in full generality. In the following we will try to convey an intuition for the threshold formula.

5. r -matched graph r=2r=2 Graph r -matched Graphs Random graph Random r -matched graph - generate G n, m - randomly partition the m edges into sets of size r a family of disjoint r -sets of edges. - have well-defined notion of subgraph inclusion - Achlioptas process after N steps is distributed as

6. The Gluing Intuition ‚ gluing together ’ E [ # copies of in ] F ³ E [ # copies of F in G n, m ] Threshold for „ contains “ = Threshold for „ G n, m contains F “ ³ n v(F) (m/n 2 ) e(F)

7. Bollobás (1981): Threshold for the appearance of F in G n, m is given by m=m(n)m=m(n) The Gluing Intuition Exampl e: F =, r = 2 Greedy strategy: e / v = 5/4 As long as a.a.s. this subgraph does not appear, and hence we do not lose. Our approach: view Achlioptas process as ‘static’ object and use (the r -matched version of) the above theorem.

8. 1 st Observation: Subgraph Sequences F =, r = 2 e / v = 11/8 = 1.375 Greedy strategy: 0 Maximization over a sequence of subgraphs of F e / v = 15/10 = 1.5 Optimal strategy: As long as a.a.s. this subgraph does not appear, and hence we do not lose.

9. 2 nd Observation: Edge Orderings Ordered graph: pair oldest edge youngest edge 2 3 7 4 6 1 5 0 Minimization over all possible edge orderings of F 1 2 2 e / v = 19/14 = 1.357... 2 F =, r = 2 Edge ordering ¼ 1 : 1 2 Optimal Strategy for ¼ 1 : 1 2 2 1 2 e / v = 17/12 = 1.417... Edge ordering ¼ 2 : 1 2 1 2 2 Optimal Strategy for ¼ 2 :

10. F 3- 4 4 4 4 F 2- 3 4 F 1- 3 r-1 4 Calculating the Threshold Minimize over all possible edge orderings ¼ of F Maximize e ( J )/ v ( J ) over all subgraphs J 4 F ¼ r 2 3 4 F 2 3 4 F 1 F ¼F ¼ Maximization over a sequence of subgraphs of F 5 J H1H1 H2H2 H3H3 H1H1 H2H2 H3H3 H3H3 H3H3

11. Calculating the Threshold (Example) 7 F 6- … F ¼F ¼ 1 3 4 6 2 5 3 4 6 5 2 7 7 F =, r = 2 Minimize over all possible edge orderings ¼ of F Maximize e ( J )/ v ( J ) over all subgraphs J 4 F ¼ 5 3 7 4 6 5 3 7 4 6 5 7 7 7 7 J e ( J )/ v ( J ) = 19/14 Maximization over a sequence of subgraphs of F 1 3 7 4 6 2 5 F ( F, ¼ ) 2 3 7 4 6 5 F 1- 3 7 4 6 5 F 2-

12. Maximization over a sequence of subgraphs of F Our Result ( explained) Theorem: Let F be any fixed graph and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is where Minimization over all possible edge orderings of F Maximize e ( J )/ v ( J ) over all subgraphs J 4 F ¼

13. A Canonical Strategy In order to avoid copies of F, we want to keep the number of copies of all (ordered!) subgraphs ( H, ¼ ) of F as low as possible. For N = n 2 – µ fixed: The problem has a recursive structure. We can compute a canonical strategy and values ¸ µ ( H, ¼ ) such that in N = n 2 – µ many steps, at most copies of ( H, ¼ ) are created (for all subgraphs ( H, ¼ ) simultaneously!) In order to avoid copies of F we use this canonical strategy with Alternatively, the threshold can be defined directly via the recursion.

14. About the ¸ -values 2 1 harmless dangerous F =, r = 2 ¸ µ (H, ¼) is the exponent of the expected number of copies of a ‚typical‘ history graph J of (H, ¼),

15. A Canonical Strategy For each edge calculate the level of danger it entails as the most dangerous (ordered) subgraph this edge would close Among all edges, pick the least dangerous one f1f1 f2f2 frfr … is more dangerous than 5

16. Lower Bound Proof Deterministic Lemma: By our strategy, each black copy of some ordered graph is contained in a copy of a grey-black r -matched graph H ’ with expectation at most... 1 2 H’H’ “History graph” If , each possible history does a.a.s. not appear in. Constantly many histories H ’ of ending up with a copy of F a.a.s. no copy of F is created. Technical work! F = r = 2 2 1 1 2 harmless dangerous This might be the same edge “Bastard”

17. Goal: force a copy of F. We fix the most dangerous ordering ¼ and force a copy of (the r-matched version of) F that contains a copy of F. This is enforced if we get r edges all closing the central copy of a copy of. We do this inductively in e(F) rounds, doing a small subgraphs type variance calculation in each round. works out if. Upper Bound Proof 4 4 3 4 2 3 4 1 2 3 4 4 4 3 4 F ¼F ¼ F

18. Conclusion The problem can be solved completely using the methods of first and second moment only. The notion of r -matched graphs allows a combinatorial interpretation of the threshold formula. Open questions: (ongoing work with Michael Krivelevich) What if r = r ( n )  1 ? What if we want to create a copy of F as quickly as possible?

19. Thank you! Questions?