Online Ramsey Games in Random Graphs

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Online Ramsey Games in Random Graphs

• Reto Spöhel, ETH Zürich
• Joint work with Martin Marciniszyn and Angelika
  Steger

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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)

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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.

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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)

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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.

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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.?

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Example
F K3, r 2
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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.,

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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.

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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

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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)

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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

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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.

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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

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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.

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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
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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
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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
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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
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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

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

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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.
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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).

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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

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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.

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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

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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?

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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

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• Thank you!
• Questions?

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