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Online Ramsey Games in Random Graphs

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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

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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
  • 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, 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
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