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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
4
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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7
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
27
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…
15
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
2323
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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