Online Vertex-Coloring Games in Random Graphs

Reto Spöhel

(joint work with Martin Marciniszyn; appeared at SODA ’07)

Introduction• Ramsey theory: when are the edges/vertices of

a graph colorable with r colors without creating a monochromatic copy of some fixed graph F ?

•We call such colorings valid (with respect to F ).

Introduction• Ramsey theory: when are the edges/vertices of

a graph colorable with r colors without creating a monochromatic copy of some fixed graph F ?

• For random graphs: solved in full generality by

•[Łuczak, Ruciński, Voigt (1992)] (vertex colorings)

•[Rödl, Ruciński (1995)] (edge colorings)

Introduction• Ramsey theory: when are the edges/vertices of a graph

colorable with r colors without creating a monochromatic copy of some fixed graph F ?

• For random graphs: solved in full generality by

•[Łuczak, Ruciński, Voigt (1992)] (vertex colorings)

•[Rödl, Ruciński (1995)] (edge colorings)

• Throughout this talk: Gn, p denotes the random graph on n vertices obtained by including each possible edge with probability p = p(n) independently.

• We consider the vertex-coloring case

Introduction

• [Łuczak, Ruciński, Voigt (1992)]: For any fixed graph F and any fixed number of colors r ¸ 2, there are explicit threshold functions p0(F, r, n) such that

•In fact, p0(F , r, n) = p0(F , n), i.e., the threshold does not depend on the number of colors r

•e.g., p0(K3, 2, n)= p0(K3, 1000, n)= n-2/3

• We transfer this result into an online setting, where the vertices of Gn, p have to be colored one by one before seeing the entire graph.

The online vertex-coloring game

• Rules:• random graph Gn, p , initially hidden

• vertices are revealed one by one along with induced edges

• vertices have to be instantly (‚online‘) colored with one of r R 2 available colors.

• game ends when monochromatic copy of some fixed forbidden graph F appears

• Question:

• How dense can the underlying random graph be such that Painter can color all vertices a.a.s.?

Example

F = K3, r = 2

Main result (simplified)

• Theorem (Marciniszyn, S.; SODA ’07)Let F be a clique or a cycle of arbitrary size.

Then the threshold for the online vertex-coloring game with respect to F and with r R 2 available colors is

i.e.,

Bounds from ‚offline‘ graph properties

• Gn, p contains no copy of F

Painter wins with any strategy

• Gn, p allows no r-vertex-coloring avoiding F Painter loses with any strategy

the thresholds of these two ‚offline‘ graph properties bound p0(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

‚Gn, p contains a copy of F‘

is

where

Appearance of small subgraphs

• m(F) is half of the average degree of the densest subgraph of F.

• For ‚nice‘ graphs – e.g. for cliques or cycles – we have

(such graphs are called balanced)

Vertex-colorings of random graphs

• Theorem (Łuczak, Ruciński, Voigt, 1992)Let F be a graph and let r R 2.The threshold for the graph property

‚every r-vertex-coloring of Gn, p contains a monochromatic copy of F‘

is

where

Vertex-colorings of random graphs

• For ‚nice‘ graphs – e.g. for cliques or cycles – we have

(such graphs are called 1-balanced)

• If F is 1-balanced, is also the threshold for the property

‚There are more than n copies of F in Gn, p ‘

• Intuition: For p [ p0 , the copies of F overlap in vertices, and coloring Gn, p becomes difficult.

• For arbitrary F and r we thus have

• Theorem Let F be a clique or a cycle of arbitrary size.

Then the threshold for the online vertex-coloring game with respect to F and with r R 1 available colors is

• r = 1 Small Subgraphs

• r exponent tends to exponent for offline case

Main result revisited

Lower bound (r = 2)

• Let p(n)/p0(F, 2, n) be given. We need to show:• There is a strategy which allows Painter to color

all vertices of Gn, p a.a.s.

Lower bound (r = 2)

• We consider the greedy strategy: color all vertices red if feasible, blue otherwise.

after the losing move, Gn, p contains a blue copy of F, every vertex of which would close a red copy of F.

• For F = K4, e.g. or

Lower bound (r = 2)

Painter is safe if Gn, p contains no such ‚dangerous‘ graphs.

• LemmaAmong all dangerous graphs, F * is the one with minimal average degree, i.e., m(F *) % m(D) for all dangerous graphs D.

F *

D

Lower bound (r = 2)

• CorollaryLet F be a clique or a cycle of arbitrary size.Playing greedily, Painter a.a.s. wins the online vertex-coloring game w.r.t. F and with two available colors if

F *

Lower bound (r = 3)

• CorollaryLet F be a clique or a cycle of arbitrary size.Playing greedily, Painter a.a.s. wins the online vertex-coloring game w.r.t. F and with three available colors if

F 3*F *

Lower bound

• CorollaryLet F be a clique or a cycle of arbitrary size.Playing greedily, Painter a.a.s. wins the online vertex-coloring game w.r.t. F and with r R 2 available colors if

…

The general case

• In general, it is smarter to greedily avoid a suitably chosen subgraph H of F instead of F itself.

general threshold function for game with r colors is

where

• Maximization over r possibly different subgraphs Hi F, corresponding to a „smart greedy“ strategy.

F

H

A surprising example

F = H1 ] H2

H1 H2

(lower bound only)

Upper bound

• Let p(n)[p0(F, r, n) be given. We need to show:

• The probability that Painter can color all vertices of Gn, p tends to 0 as n , regardless of her strategy.

• Proof strategy: two-round exposure & induction on r

•First round•n/2 vertices, Painter may see them all at once

•use known offline results

•Second round•remaining n/2 vertices

•Due to coloring of first round, for many vertices one color is excluded induction.

Upper bound

V1 V2

F °

1) Suppose Painter‘s offline-coloring of V1 creates many (w.l.o.g.) red copies of F °

2) Depending on the edges between V1 and V2, these copies induce a set Base(R) 4 V2 of vertices that cannot be colored red.

3) Edges between vertices of Base(R) are independent of 1) and 2)

Base(R) induces a binomial random graph

Base(R)

F

need to show: Base(R) is large enough for induction hypothesis to be applicable.

• There are a.a.s. many monochromatic copies of F‘° in V1 provided that

• work (Janson, Chernoff, ...) These induce enough vertices in (w.l.o.g.)

Base(R) such that the induction hypothesis is applicable to the binomial random graph induced by Base(R).

Upper bound

Main result (full)

• Theorem (Marciniszyn, S.; SODA ’07)Let F be a graph for which at least one F ° satisfies

Then the threshold for the online vertex-coloring game w.r.t. F and with r R 1 colors is

• This threshold formula is not true for arbitrary graphs F!

F °

Intermission…(Questions?)

The online edge-coloring game

• Rules:• one player, called Painter

• start with empty graph on n vertices

• edges appear u.a.r. one by one and have to be colored instantly (‚online‘) either red or blue

• game ends when monochromatic triangle appears

• Question: How many edges can Painter color?

• Theorem (Friedgut, Kohayakawa, Rödl, Ruciński, Tetali, 2003):

The threshold for this game is N0(n)= n4/3.

(easy, not main result of paper)

Back to online edge colorings

• Threshold is given by appearance of F *, yields threshold formula similarly to vertex case.

• Lower bound:

• Much harder to deal with overlapping outer copies!

• Works for arbitrary number of colors.

• Upper bound:

• Two-round exposure as in vertex case

• But: unclear how to setup an inductiveargument to deal with r R 3 colors.

F_F °

?6F *

Online edge colorings

• Theorem (Marciniszyn, S., Steger, 2009)Let F be a graph that is not a tree, for which at least one F_ satisfies

Then the threshold for the online edge-coloring game w.r.t. F and with two colors is

F_