Online Vertex Colorings of Random Graphs Without Monochromatic SubgraphsReto Spöhel, ETH ZurichJoint work with Martin Marciniszyn

Introduction

• Chromatic Number: Minimum number of colors needed to color vertices of a graph such that no two adjacent vertices have the same color.

• Generalization: Instead of monochromatic edges, forbid monochromatic copies of some other fixed graph F.

• Question: When are the 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 Luczak, Rucinski, Voigt, 1992

F = K3, r = 2

Introduction

• ‚solved in full generality‘: Explicit threshold functionp0(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 (!)

• The threshold behaviour is even sharper than shown here.

• We transfer this result into an online setting, where the vertices of Gn, p have to be colored one by one, without seeing the entire graph.

Introduction: our results

• Explicit threshold functions p0(F , r, n) for online-colorability with r R 2 colors for a large class of forbidden graphs F , including cliques and cycles of arbitrary size.

• Unlike in the offline case, these thresholds

•depend on the number of colors r

•are coarse.

Introduction: related work

• Question first considered for the analogous online edge-coloring (‚Ramsey‘) problem•Friedgut, Kohayakawa, Rödl, Rucinski,

Tetali, 2003: F = K3, r = 2

•Marciniszyn, S., Steger, 2005+: F e.g. a clique or a cycle, r = 2

• Theory similar for edge- and vertex-colorings, but edge case is considerably more involved.

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 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) 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

Generalization

• 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.

• Proved as a lower bound in full generality.

• Proved as an upper bound assuming

Thank you! Questions?

Similarly: 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 ³ 3 colors.

F*

F_F°

?6

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_

Online vertex colorings

• Theorem (Marciniszyn, S., 2006+)Let 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 *