# Online Ramsey Games in Random Graphs - PowerPoint PPT Presentation

PPT – Online Ramsey Games in Random Graphs PowerPoint presentation | free to download - id: 1d323d-ZDc1Z

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Online Ramsey Games in Random Graphs

Description:

### Then the threshold for the online vertex-coloring game with respect to F and ... or. Lower bound (r = 2) Painter is safe if Gn, p contains no such 'dangerous ... – PowerPoint PPT presentation

Number of Views:32
Avg rating:3.0/5.0
Slides: 33
Provided by: reto
Category:
Tags:
Transcript and Presenter's Notes

Title: Online Ramsey Games in Random Graphs

1
Online Ramsey Games in Random Graphs
• Reto Spöhel, ETH Zürich
• Joint work with Martin Marciniszyn and Angelika
Steger

2
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
• Luczak/Rucinski/Voigt, 1992 (vertex colorings)
• Rödl/Rucinski, 1995 (edge colorings)

3
Introduction
• solved in full generality Explicit threshold
functionsp0(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 except
• The threshold behaviour is even sharper than
shown here except
• We transfer these results into an online setting,
where the edges/vertices of Gn, p have to be
colored one by one, without seeing the entire
graph.

4
The online edge-coloring game
• Rules
• one player, called Painter
• 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, Rucinski,
Tetali, 2003)
• The threshold for this game is N0(n) n4/3.
• (easy, not main result of paper)

5
Our results
• Online edge-colorings
• threshold for online-colorability with 2 colors
for a large class of graphs F including cliques
and cycles
• Online vertex-colorings main focus of this
talk
• threshold for online-colorability with r R 2
colors for a large class of graphs F including
cliques and cycles
• Unlike in the offline cases, these thresholds are
coarse and depend on the number of colors r.

6
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.?

7
Example
F K3, r 2
8
Main result (simplified)
• 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.,

9
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.

10
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

11
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)

12
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

13
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.

14
Main result revisited
• 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

15
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.

16
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
17
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.

D
F
18
Lower bound (r 2)
• Corollary
• Let 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
19
Lower bound (r 3)
• Corollary
• Let 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
F 3
20
Lower bound
• Corollary
• Let 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

21
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.

H
F
22
A surprising example
F H1 H2
H1
H2
(lower bound only)
23
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.

24
Upper bound
F
F
• Suppose Painters offline-coloring of V1 creates
many (w.l.o.g.) red copies of F
• Depending on the edges between V1 and V2, these
copies induce a set Base(R) 4 V2 of vertices that
cannot be colored red.
• Edges between vertices of Base(R) are independent
of 1) and 2)
• ? Base(R) induces a binomial random graph

Base(R)
V2
• V1

? need to show Base(R) is large enough for
induction hypothesis to be applicable.
25
Upper bound
• 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).

26
Main result (full)
• Theorem (Marciniszyn, S., 2006)
• 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

27
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.

28
Online edge colorings
• Theorem (Marciniszyn, S., Steger, 2005)
• 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

29
Open problems
• More colors (edge case).
• Simplest open case F K3, r 3
• General graphs (vertex and edge case)
• is not the truth in
general!
• Is there an explicit general threshold formula?

30
Outlook balanced online games
• In every step, r vertices appear at once, and
Painter has to assign each of the r available
colors to exactly one of these vertices.
• Prakash, S. (2007)
• Threshold result for a large class of graphs F
and arbitrary number r of colors.
• The threshold is strictly lower than the
unbalanced online threshold, but satisfies an
analogous convergence.
• The edge case was studied by Marciniszyn,
Mitsche, Stojakovic (2005)
• Threshold result for F e.g. a cycle and two
colors.
• We hope to improve on this in the near future

31
• Thank you!
• Questions?

32
(No Transcript)