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Online Graph Avoidance Games in Random Graphs
Reto SpöhelDiploma ThesisSupervisors: Martin Marciniszyn, Angelika Steger
Outline
• The online graph avoidance game•Rules and known result
•Main result
• Proof•Lower bound
•Upper bound
• Outlook•The game with more colors
The online graph avoidance game
• Rules:
•one player, called the Painter
•starts with the empty graph on n vertices
•edges appear one by one u.a.r. and have to be instantly (‚online‘) colored either red or blue
•The game ends as soon as the Painter closes a monochromatic copy of a fixed forbidden graph F.
• Question:
•How many edges can the Painter typically color?
Known result
• Theorem (Friedgut et al., 2003)The threshold for the online triangle avoidance game with two colors is
i.e.,
Main result
• Theorem (Main result for cliques)Let F be a clique of arbitrary size.Then the threshold for the online F-avoidance game with two colors is
Bounds from ‚offline‘ graph properties
• Let G(n, N) denote the graph on n vertices obtained by inserting N edges one by one, u.a.r.
• G(n, N) is distributed uniformly over all graphs with n vertices and N edges.
• Game ends at time N:•G(n, N) contains a copy of F .
•G(n, N - 1) can be 2-coloured avoiding monochromatic copies of F .
• ) the thresholds of these two ‚offline‘ graph properties bound N0(n) from below and above.
Appearance of small subgraphs
• Theorem (Bollobás, 1981)Let F be a non-empty graph.The threshold for the graph property
‚G(n,N) contains a copy of F‘is
where
Appearance of small subgraphs
• m(F) is the edge density of the densest subgraph of F.
• For ‚nice‘ graphs – e.g. for cliques – we have
(such graphs are called balanced)
Ramsey theory in random graphs
• Theorem (Rödl/Rucinski, 1995)Let F be a graph which is not a star forest.The threshold for the graph property
‚every 2-coloring of G(n,N) contains a monochromatic copy of F‘
is
where
Ramsey theory in random graphs
• For ‚nice‘ graphs – e.g. for cliques – we have
(such graphs are called 2-balanced)
• . is also the threshold for the graph property‚There are more copies of F than edges in G(n,N)‘
• Intuition: For N À N0 this forces the copies of F to overlap substantially, and coloring G(n,N) becomes difficult.
Main result revisited
• For arbitrary F we thus have
• Theorem (Main result for cliques)Let F be a clique of arbitrary size. Then the threshold for the online F-avoidance game with two colors is
Lower bound
• Let N(n) ¿ N0(n) be arbitrary. We need to show:
•There is a strategy which allows the Painter to color N(n) edges with probability tending to 1 as n!1.
• We consider the greedy strategy: color all edges red if feasible, blue otherwise.
• Proof strategy:•Reduce the event that the Painter fails to
the appearance of a certain dangerous graph F * in G(n,N).
•Apply ‘small subgraphs’ theorem.
[‘asymptotically almost surely, a.a.s.’]
Lower bound
• Analysis of the greedy strategy:•color all edges red if feasible, blue
otherwise.
• ) after the losing move, the graph contains a blue copy of F, every edge of which would close a red copy of F.
•For F=K4, e.g. or
Lower bound
• ) A greedy Painter loses if the edges of one of these dangerous graphs appear in a bad order.
• ) The Painter is secure as long as none of these graphs appear in G(n,N).
Lower bound
• LemmaF * is a.a.s. the first dangerous graph which appears in G(n,N).
• Proof idea:• By the ‚small subgraphs‘ theorem, we need to
show m(F *) < m(D) for all other dangerous graphs D.
DF *
Lower bound
• Construct a given dangerous graph D inductively by merging edges and vertices of F *…
D…and use amortized analysis to prove that m(D) > m(F *).
F *D
Lower bound
• Corollary (Lower bound)Let F be a clique of arbitrary size.Playing greedily, the Painter can a.a.s. color any
edges.
F *
Upper bound
• Let N(n) À N0(n) be arbitrary. We need to show:
•For every strategy of the Painter, the probability that she can color N(n) edges tends to 0 as n!1.
Upper bound
• Proof strategy: two-round exposure•First round
•N0 edges, Painter may see them all at once
•a.a.s. every coloring creates many ‘threats‘.
•use results from (offline) Ramsey theory
•Second round
•remaining N1 À N0 edges
•the Painter will a.a.s. encounter a threat and hence lose the game
•use second moment method
• Consider G(n, N0) = R [ B, the 2-coloring assigned to the first N0 edges by the Painter.
• Base(R) := {edges which would close a red copy of F }
• All edges in Base(R) have to be colored blue if presented to the Painter.
• ) Painter loses in second round as soon as she is given a copy of F ½ Base(R).
Second round: the base graph
Second round: threats
• Threats = copies of F in Base(R) or Base(B)
• If there are many [=:M] threats after the first round, by the second moment method a.a.s one of them is hit in the second round.•many: enough to ensure that [X] ! 1, where X:=
number of threats hit in second round.
•second moment method: Var[X] ! 0 fast enough to guarantee that X is a.a.s. close to [X].
• Threats = copies of F ½ Base(R) are induced by copies of ½ R.
• ) We want to find many copies of in either R or B.
First round: looking for threats
A counting version
• Theorem (Rödl/Rucinski, 1995)Let H be any non-empty graph, and let
Then there is a constant c = c(H) > 0 such that a.a.s every 2-coloring of G(n,N) contains at least monochromatic copies of H.
• We will apply this with H = and N=N0 to find many monochromatic copies of in G(n, N0).
• ) There are a.a.s. M monochromatic copies of in G(n,N0), provided that
• These induce M threats ) a.a.s. the Painter loses in the second round.
First round: finding the threats
Upper bound
• Corollary (upper bound) Let F be a clique of arbitrary size.Regardless of her strategy, the Painter is a.a.s not able to colour any
edges.
Main result
• Theorem (Main result)Let F be a 2-balanced and regular graph for which at least one satisfies
Then the threshold for the online F-avoidance game with two colors is
Special Cases
• Corollary (Clique avoidance games)For l ¸ 2, the threshold for the online Kl-avoidance game with two colors is
• Corollary (Cycle avoidance games) For l ¸ 3, the threshold for the online Cl-avoidance game with two colors is
Outlook: the game with more colors
• Same rules, but Painter now has r ¸ 2 colors available.
• There is still an obvious greedy strategy:•number the colors from 1 to r
•always use the lowest number color which does not close a monochromatic copy of F.
• F e.g. a clique ) there is a unique ‚basic dangerous graph‘
Outlook: the game with more colors
• If is the first dangerous graph which appears in G(n,N), the greedy strategy ensures a.a.s. survival up to any
edges.
Outlook: the game with more colors
• We believe:For l ¸ 2 and r ¸ 1, the threshold for the online Kl-avoidance game with r colors is
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