Online Ramsey Games in Random Graphs

Reto SpöhelJoint work with Martin Marciniszyn and Angelika Steger

Introduction

• Ramsey theory: when are the vertices/edges 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

• Luczak/Rucinski/Voigt, 1992 (vertex colorings)

• Rödl/Rucinski, 1995 (edge colorings)

Introduction

• ‚solved in full generality‘: There is a threshold functionp0(F , r, n) such that

• In fact, p0(F , r, n) = p0(F , n) in almost all cases, i.e., the threshold does not depend on the number of colors r (in order of magnitude).

• We transfer these results into an online setting, where vertices/edges have to be colored one by one, without seeing the entire graph.

Overview of results

• Online vertex-colorings (this talk):

• threshold for online-colorability with r R 2 colors for a large class of graphs F including cliques and cycles

• Online edge-colorings:

• threshold for online-colorability with 2 colors for a large class of graphs F including cliques and cycles

•only non-trivial lower bounds for more than 2 colors.

• Unlike in the offline case, these thresholds depend on the number of colors r.

The online vertex-coloring game

• Rules:

• one player, called Painter

• 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 as soon as Painter closes a monochromatic copy of some fixed forbidden graph F.

• 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

• Theorem (Marciniszyn/S., 2006)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 (Luczak/Rucinski/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)

• . 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.

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

• Proof strategy:• reduce the event that Painter fails to the

appearance of a certain dangerous graph F * in Gn, p .

• apply Small Subgraphs Theorem.

Lower bound (r = 2)

• Analysis of 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

…

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 results from (offline) Ramsey theory

•Second round•remaining n/2 vertices

•Further restrictions due to coloring of first round

•For many vertices one color excluded induction.

Upper bound

V1 V2

F °

1) 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 revisited

• TheoremLet F be a 1-balanced 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

F °

F *

Outlook

• 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

• Proved as a lower bound in full generality

• Proved as an upper bound assuming

Thank you! Questions?

Online edge colorings

• Theorem (Marciniszyn/S./Steger, 2005)Let F be a 2-balanced 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 *

F_