1. 1 Hamiltonicity Games Michael Krivelevich Tel Aviv University

2. 22 Hamiltonocity game - definition Def.: A Hamilton cycle (HC) in a graph G is a cycle passing through all vertices of G Hamiltonicity game: -Played on the edges of the complete graph K n -Two players: Player 1, Player 2 (for now)‏ -Take turns in claiming one unoccupied edge each -Player 2 (usually) starts -Player 1 wins if completes a HC by the end, Player 2 wins otherwise (notice the asymmetric roles of players)

3. 33 Chvátal-Erdős paper - Players are now: Maker and Breaker -Breaker takes b≥1 unoccupied edges at a time (game bias)‏ Notation: H(G,b) := 1:b Maker-Breaker Hamiltonicity game played on E(G); Maker wins iff creates a HC in the end CE: considered the case G=K n

4. 4 Chvátal-Erdős’ result Th.: The unbiased Hamiltonicity game H(K n,1) on the complete graph K n is Maker’s win, for all large enough n. Proof sketch: Stage 1 (n/3 rounds): M creates a cycle C 0, n/4 ≤ |C 0 | ≤ n/3; Stage 2 (O(1) rounds): M absorbs high degree vertices of B into his cycle; Stage 3 (2 rounds per vertex): M absorbs all remaining vertices into the cycle Altogether: ≤ 2n rounds 4 v C

5. 5 Open questions -“suff.large n” – how large should n be? -“ M wins in ≤ 2n moves ” – how fast is M guaranteed to win? -“ It is not unlikely ” that M can win against bias b(n)→∞: -What happens for b≥1? -What is the largest bias b=b(n) for which M still wins? 5

6. 6 Open questions (cont.)‏ -How about boards G other than K n ? What is the sparsest board G where M still wins? -Who is typically the winner on a random board? -Other game types (Avoider-Enforcer games)‏ 6

7. 7 How large should n be? -rather technical/minor (usually think of n→∞)‏ Papaioannou’82: Th.: n≥600 – Maker wins H(K n,1)‏ Conj.: min { n: H(K n,1) is M’s win } = 8 Hefetz, Stich’09: Th.: n≥38 – Maker wins (as 2 nd player)‏ n≥29 – Maker wins (as 1 st player)‏ n=8 – Breaker wins (disproving Papaioannou’s conj.) 7

8. 8 How fast? Notation: τ(n)=τ(H(K n,1)) := min. number of moves during which Maker is guaranteed to win = move number of the game (J. Beck)‏ CE’78: τ(n)<2n Obviously: τ(n) ≥ n+1 (After first n-1 moves M has ≤1 threat – which B can block)‏ Hefetz, K., Stojakovič, Szabó’09: τ(n)≤n+2 τ(Hamilton path game on K n )=n-1 - optimal 8

9. 9 How fast? (cont.)‏ Proof idea: Stage 1: M builds a perfect matching + an edge lasts n/2+1 rounds Stage 2: lasts n/2-2 rounds M gradually (and carefully) merges his paths into a single path = Hamilton path Stage 3: 3 more rounds M uses Pósa’s rotation, creates a double trap 9 P v1v1 vivi v i+1 vnvn P’ v1v1 vivi v i+1 vnvn

10. 10 How fast? (cont.)‏ Th. (Hefetz, Stich’09): τ(n)=n+1, for large enough n. Proof:Similar to HKSS, but much more involved technically -13pp of careful case analysis… Questions of similar type (HKSS):  min degree 1?  perfect matching?  spanning k-connectivity? 10

11. 11 Playing against bias Recoup: Maker-Breaker, played on E(K n )‏ Breaker starts Maker: one edge each time Breaker: b≥1 edges each time b=1 – all too easy for Maker… ?Who wins for a given b? ?What is the largest b*=b*(n) for which Maker still wins? Is it true that b*(n) →∞? 11

12. 12 Critical bias Monotonicity: Easy to see: H(G,b) is M’s win  H(G,b-1) is M’s win as well Critical bias: b*:=b*(n)=max{b: H(K n,b) is Maker’s win} Q.: Determine/approximate b*(n)? 12 3 21b* bias MM M MBBB Critical point: game changes hands M

13. 13 Overcoming the bias Bollobás, Papaionnau’82: b*≥, for some c>0 quantum leap… Beck’85: b=, (c= >0)  Maker still wins… (Proof: expander/rotation-extension techniques…)‏ 13

14. 14 Where should the critical bias be? M-B, on E(K n ), 1:b Th. (CE’78): b≥  Breaker can isolate a vertex  Wins H(K n,b) (and lots of other games)‏ Proof: B builds an isolated (from M’s edges) clique K, |K|= ; then isolates one of the vertices of K (BoxGame)‏ 14

15. 15 Where should the critical bias be? (cont.)‏ Random graph intuition (Erdős paradigm) – appeared in CE too  Conj.: b*(n)= (i.e., CE’s isolate a vertex bound is asymptotically tight)‏ -Made it to the list of 7 most humiliating open problems of Beck… 15

16. 16 Indeed! K.,10+: “The critical bias for the Hamiltonicity game is (1+o(1))n/lnn” Proof idea: builds heavily on Gebauer, Szabó’09 GS: b≤  Maker can build a graph of min. degree 1 -Actually of min. degree = const; -Actually can complete constant # of edges at v before Breaker gets (1-δ)n egdes at v Proof: greedy strategy/potential function 16

17. 17 Indeed!(cont.)‏ K.: Twist M’s strategy: touch the same vertex as in GS, but choose a random incident edge! (Random strategy wins with positive probability  exists a deterministic winning strategy)‏  Maker can create a spanning expander in Θ(n) moves – Stage 1 -turns it into a connected graph in O(1) steps – Stage 2 -brings it to Hamiltonicity (Pósa’s lemma) in ≤n more steps – Stage 3 At each step Θ(n 2 ) ways to make a longest path in M’s graph longer/ to close a HC – can’t all be blocked by Breaker 17

18. 18 Finding the critical bias in biased M-B games -most important meta-question for biased games  min. degree 1 CE’78; GS’09  connectivity CE’78; GS’09  creating a copy of a fixed graph HCE’78; Bednarska,Łuczak’00  further properties? 18

19. 19 Playing on other/sparse boards Recall: game H(G,b) :1:b, Maker-Breaker played on the edge set of a graph G Maker wins iff constructs a HC in the end Here: mostly case b=1 (unbiased games)‏ Q. : What is the sparsest board G on n vertices where Maker wins? Formally: := min { |E(G)|: G=(V,E), |V|=n, Maker wins H(G,1)} (Related to size Ramsey numbers)‏ 19

20. 20 Sparsest winning board Clearly: should require δ(G)≥4 (otherwise B takes all but one edge incident to a min degree vertex)‏  ≥ 2n Upper bound? Constant max. degree graph G where M wins? Th. (HKSS’10+): LB: prove: # of vertices of degree ≥ 6 in G ≥ number of vertices of degree 4 20

21. 21 Sparsest winning board (cont.)‏ UB: G = K i = constant (=40) size clique, t →∞ 21 K0K0 K1K1 K t-1 KiKi u t-1 u0u0 u i-1 uiui

22. 22 Sparsest winning board (cont.)‏ i-th fragment: M can get:Hamilton cycle inside K i edges from both u i-1,u i to K i  A Hamilton path from u i-1 thru K i to u i Gluing these fragments  a HC in G. ■ Other questions of the same type (HKSS): -min. degree k; -spanning k-connectivity; -making a given bounded degree spanning tree T. 22 KiKi uiui u i-1 uiui

23. 23 Playing on a random board Random graph G(n,p) : V={1,…,n}=[n] 1≤i≠j≤n: Pr[(i,j)  E(G)]=p=p(n), indep. Q.: For a given p=p(n), who typically wins for a board G~G(n,p)? min{p=p(n): M typically wins H(G,1) for G~G(n,p)}? (typically = with prob.→1 as n→∞)‏

24. 24 Playing on a random board (cont.)‏ What’s the guess? For M to hope to win: -G should contain a Hamilton cycle; -δ(G)≥4. Luckily for G(n,p) happen ≈ at the same time Th. (Komlós-Szemerédi’83; Bollobás’84):  with high prob.(whp) G~G(n,p) is Hamiltonian Prop.  with high prob. δ(G(n,p))≥4

25. 25 Playing on a random board (cont.)‏ TONCAS: Th. (Ben-Shimon, Ferber, Hefetz, K.’10+):  G~G(n,p) is whp such that Maker wins H(G,1)‏ (in fact proved the hitting time version of this result)‏ Also: Th. (Ben-Shimon, K., Sudakov’10+): Consider the random d-regular graph G n,d. Then there is abs. const d 0 s.t. d≥d 0  G~G n,d is whp s.t. that M wins H(G,1)‏ Moral: not only G(n,p)/G n,d is whp Hamiltonian, it is robustly Hamiltonian – Maker can construct a HC against adversarial B.

26. 26 Playing on a random board (cont.)‏ Proof: delicate application of Pósa’s rotation-extension technique + new tools (advancement to Hamiltonicity should be robust to withstand adversarial Breaker)‏ Similar questions: Board = G(n,p) (or G n,d or some other random graph model)‏ M-B, 1:1 (or 1:b in general)‏ min p=p(n) for Maker to create whp: -a perfect matching (Stojakovič, Szabó’05; BFHK’10+); -a k-connected spanning subgraph (SS’05; BFHK’10+); -a copy of a fixed graph H (SS’05); -other games?

27. 27 Bending the rules Instead of Maker-Breaker (Maker wins if completes a HC in the end; Breaker wins otherwise)‏ -other game rules? -Strong games: both players are trying to achieve Hamilt’ty; whoever builds a HC first wins, otherwise a draw -essentially nothing is known; -should be probably very hard. -Misére version?

28. 28 Avoider-Enforcer games -played on the edges of K n (can generalize to arbitrary G)‏ -two players: Avoider, Enforcer -Avoider takes at least one edge at a time -Enforcer takes at least b edges at a time P = monotone graph property (say, Hamiltonicity)‏ Avoider wins is avoids having P in the end Enforcer wins otherwise (=forces Avoider to have P)‏ Changing the more natural rule (=exactly); guarantees bias monotonicity – appears the right thing to do

29. 29 Avoider-Enforcer Hamiltonicity games -Featured in the “humiliating list” of Beck Th. (K., Szabó’08): b≤  Enforcer wins the 1:b A-E Hamilt’y game on E(K n )‏ Th. (HKSS’10): b≥  Avoider can avoid touching every vertex in 1:b A-E game on E(K n )  wins the A-E Hamiltonicity game Conclusion: b*(A-E Hamiltonicity)=

30. 30 Avoider-Enforcer games Open questions to consider: -A-E Hamiltonicity game under traditional rules? -A-E avoiding small graphs game? -just about anything else?