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Hamiltonicity Games

Michael Krivelevich

Tel Aviv University

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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 Kn

- 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)

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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=Kn

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Chvátal-Erdős’ result

Th.: The unbiased Hamiltonicity game H(Kn,1) on the complete graph Kn is Maker’s win, for all large enough n.

Proof sketch:Stage 1 (n/3 rounds): M creates a cycle C0, n/4 ≤ |C0 | ≤ 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

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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

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Open questions (cont.)

- How about boards G other than Kn?

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)

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How large should n be?

- rather technical/minor

(usually think of n→∞)

Papaioannou’82: Th.: n≥600 – Maker wins H(Kn,1)

Conj.: min{n: H(Kn,1) is M’s win}= 8

Hefetz, Stich’09: Th.: n≥38 – Maker wins (as 2nd player)

n≥29 – Maker wins (as 1st player)

n=8 – Breaker wins

(disproving Papaioannou’s conj.) 7

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How fast?

Notation: τ(n)=τ(H(Kn,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 Kn)=n-1 - optimal

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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

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Pv1 vi vi+1 vn

P’v1 vi vi+1 vn

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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?

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Playing against bias

Recoup: Maker-Breaker, played on E(Kn)

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) →∞?

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Critical bias

Monotonicity:

Easy to see: H(G,b) is M’s winH(G,b-1) is M’s win as well

Critical bias: b*:=b*(n)=max{b: H(Kn,b) is Maker’s win}

Q.: Determine/approximate b*(n)?

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321 b*

biasM M M M B B B

Critical point: game changes hands

M

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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

n

cn

ln

n

nc

lnln

ln

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2ln

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Where should the critical bias be?

M-B, on E(Kn), 1:b

Th. (CE’78): b≥ Breaker can isolate a vertex

Wins H(Kn,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

n

n

ln

)1(

n

nk

ln2

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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…

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n

no

ln

))1(1(

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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

(1 )

ln

n

n

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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(n2) ways to make a longest path in M’s graph longer/

to close a HC – can’t all be blocked by Breaker

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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 H CE’78;

Bednarska,Łuczak’00

further properties?

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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)

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ˆ ( )m n

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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

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ˆ ( )m n

ˆ2.5 ( ) 21n m n n

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Sparsest winning board (cont.)

UB:

G =

Ki = constant (=40) size clique, t →∞

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K0

K1Kt-1

Ki

ut-1 u0

ui-1ui

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Sparsest winning board (cont.)i-th fragment:

M can get: Hamilton cycle inside Ki

edges from both ui-1,ui to Ki

A Hamilton path from ui-1 thru Ki to ui

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.

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Ki

uiui-1

ui-1 ui

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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→∞)

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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 timeTh. (Komlós-Szemerédi’83; Bollobás’84): with high prob.(whp) G~G(n,p) is HamiltonianProp.

with high prob. δ(G(n,p))≥4

ln ln ln ( )n n np

n

ln 3ln ln ( )n n np

n

lim ( )n

n

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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 Gn,d. Then there is abs. const d0

s.t. d≥d0

G~Gn,d is whp s.t. that M wins H(G,1)

Moral: not only G(n,p)/Gn,d is whp Hamiltonian, it is robustly Hamiltonian – Maker can construct a HC against adversarial B.

ln 3ln ln ( )n n np

n

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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 Gn,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?

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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?

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Avoider-Enforcer games

- played on the edges of Kn

(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 endEnforcer wins otherwise (=forces Avoider to have P)

Changing the more natural rule (=exactly); guarantees bias

monotonicity – appears the right thing to do

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Avoider-Enforcer Hamiltonicity games

- Featured in the “humiliating list” of BeckTh. (K., Szabó’08):

b≤ Enforcer wins the 1:b A-E Hamilt’y game on E(Kn)

Th. (HKSS’10):

b≥ Avoider can avoid touching every vertex in 1:b

A-E game on E(Kn) wins the A-E Hamiltonicity game

Conclusion: b*(A-E Hamiltonicity)=

n

n

ln

)1(

(1 )

ln

n

n

(1 (1))

ln

o n

n

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Avoider-Enforcer games

Open questions to consider:

- A-E Hamiltonicity game under traditional rules?- A-E avoiding small graphs game?

- just about anything else?