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?

Example

n = 6, F = K3

Length of game: N = 9 edges

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

A simplification

• Lemma

where := ‘F with one edge removed‘.

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‘

F = K3; r = 3; colors: yellow, red, blue

Example: the 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