Small Subgraphs in Random Graphs and the Power of Multiple Choices The Online Case Torsten Mütze, ETH Zürich Joint work with Reto Spöhel and Henning Thomas TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A

Introduction Goal: Avoid creating a copy of some fixed graph F Achlioptas process (named after Dimitris Achlioptas) : 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 the appearance of F be avoided having this freedom of choice? F =, r = 2

Introduction N 0 = N 0 ( F, r, n ) is a threshold: N 0 = N 0 ( F, r, n ) N = N ( n ) = number of steps in the Achlioptas process There is a strategy that avoids creating a copy of F with probability 1-o(1) (as n tends to infinity) 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 … r=3r=3 n … r=4r=4 n1n1 r=1r=1

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 Krivelevich et al. conjecture a general threshold formula We disprove this conjecture and solve the problem in full generality... we will develop an intuition for the threshold formula in the following

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 Achlioptas process after N steps is distributed as

Threshold for the appearance of small subgraphs in G n, m (Bollobás, 1981): m=m(n)m=m(n) The Gluing Intuition Analogue of Bollobás‘ theorem for r -matched graphs: m=m(n)m=m(n) Our key idea: relate Achlioptas process to ‘static’ object F =, r = 2 Greedy strategy: e / v = 5/4 As long as this subgraph does not appear, hence we do not lose. “Gluing Intuition” J

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

2 nd Observation: Edge Orderings Ordered graph: pair oldest edge youngest edge Minimization over all possible edge orderings of F e / v = 19/14 = F =, r = 2 Optimal Strategy for ¼ 1 : Edge ordering ¼ 1 : e / v = 17/12 = Edge ordering ¼ 2 : Optimal Strategy for ¼ 2 :

F F 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 F F 1 F ¼F ¼ Maximization over a sequence of subgraphs of F 5 J H1H1 H2H2 H3H3 H1H1 H2H2 H3H3 H3H3 H3H3

Calculating the Threshold (Example) F 1- F 2- F 6- … e ( J )/ v ( J ) = 17/12 = F ¼F ¼ J F ( F, ¼ ) 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 Maximization over a sequence of subgraphs of F

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

Densest Versus Rarest Subgraph Duality: For the subgraph of minimum expectation equals the subgraph of maximum density Goal: Finding the densest subgraph J in F ¼ J14H1J14H1 H1H1 H2H2 J24H2J24H2 F ¼F ¼ J Idea: Look at the expectation of a fixed subgraph J in G n, m for : Gluing together densest parts does NOT yield densest subgraph! Gluing together rarest parts DOES yield rarest subgraph! - J := J 1 W J 2 If we choose such that Rarest subgraph J in F ¼ can be found recursively!

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

Special Case: Forests Theorem: Let F be any forest and let r R 2 be a fixed integer. Then the threshold for avoiding F in the Achlioptas process with parameter r is where In particular, where, and paths stars

Theorem (minimum expectation version): 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 is the unique solution of and Our Result ( explained)

The F -Avoidance-Strategy (1) Measure the harmlessness of a subgraph by the parameter 2 1 harmless dangerous F =, r = 2

The F -Avoidance-Strategy (2) 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 Our strategy considers ordered subgraphs

Lower Bound Proof Lemma: By our strategy, each black copy of some ordered graph is contained in a copy of some rare grey-black r -matched graph H ’. 1 2 H’H’ “History graph” F does not appear Constantly many histories H ’ of ending up with a copy of F Below the threshold a.a.s. none of the histories H ’ appears in Technical work! F = r = harmless dangerous This might be the same edge “Bastard” rare: expectation at most implies that H ’ does not appear in with probability 1- o(1)

Goal: force a copy of F, in some fixed order regardless of the strategy Multiround approach (#rounds = #edges of F ) In each round: count how many copies evolve further, 1st+2nd MM, small subgraph variance calculation for r -matched graphs Optimize for the best upper bound Upper Bound Proof 1st round 2nd round 3rd round

Thank you! Questions?