Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn...
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![Page 1: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/1.jpg)
Online Ramsey Games in Random Graphs
Reto Spöhel, ETH ZürichJoint work with Martin Marciniszyn and Angelika Steger
![Page 2: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/2.jpg)
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
•Łuczak/Ruciński/Voigt, 1992 (vertex colorings)
•Rödl/Ruciński, 1995 (edge colorings)
![Page 3: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/3.jpg)
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.
![Page 4: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/4.jpg)
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, Ruciński, Tetali, 2003):
The threshold for this game is N0(n)= n4/3.
(easy, not main result of paper)
![Page 5: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/5.jpg)
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.
![Page 6: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/6.jpg)
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.?
![Page 7: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/7.jpg)
Example
F = K3, r = 2
![Page 8: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/8.jpg)
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.,
![Page 9: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/9.jpg)
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.
![Page 10: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/10.jpg)
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
![Page 11: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/11.jpg)
Appearance of small subgraphs
• For ‚nice‘ graphs – e.g. for cliques or cycles – we have
(such graphs are called balanced)
![Page 12: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/12.jpg)
Vertex-colorings of random graphs
• Theorem (Łuczak, Ruciński, 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
![Page 13: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/13.jpg)
Vertex-colorings of random graphs
• For ‚nice‘ graphs – e.g. for cliques or cycles – we have
(such graphs are called 1-balanced)
• For these graphs, . 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.
![Page 14: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/14.jpg)
• 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
![Page 15: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/15.jpg)
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.
![Page 16: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/16.jpg)
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
![Page 17: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/17.jpg)
Lower bound (r = 2)
Painter is safe if Gn, p contains no such ‚dangerous‘ graphs.
• LemmaAmong all dangerous graphs, F * is the sparsest one, i.e., m(F *) % m(D) for all dangerous graphs D.
F *
D
![Page 18: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/18.jpg)
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 *
![Page 19: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/19.jpg)
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 *
![Page 20: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/20.jpg)
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
…
![Page 21: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/21.jpg)
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.
F
H
![Page 22: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/22.jpg)
A surprising example
F = H1 ] H2
H1 H2
(lower bound only)
![Page 23: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/23.jpg)
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.
![Page 24: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/24.jpg)
Upper bound
V1 V2
F °
1) Suppose 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.
![Page 25: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/25.jpg)
• 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
![Page 26: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/26.jpg)
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
F °
![Page 27: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/27.jpg)
Side remark: Trees
• Greedy strategy gives lower bound of f for any tree T on vT vertices and any number r of colors.
• Theorem (Mütze, S., 2008+):For any fixed tree T and any number r of colors, the precise threshold can be found by finite computation.• For , the threshold for the online vertex-
coloring game with respect to and with two available colors is
• But: the threshold for the online vertex-coloring game with respect to and with two available colors is at least
n¡
v rT
v rT
¡ 1
n¡
vrT
vrT ¡ 1
![Page 28: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/28.jpg)
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.
F_F °
?6F *
![Page 29: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/29.jpg)
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
F_
![Page 30: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/30.jpg)
Open problems
• More colors (edge case)•Simplest open case: F = K3, r = 3
• General non-trees• is not the truth in general!
• Is there an explicit general threshold formula?
• Trees• Is it just combinatorial chaos, or is there a hidden
pattern?
![Page 31: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/31.jpg)
Outlook: balanced online games
• The greedy strategy produces very unbalanced colorings.
• consider new ‚balanced‘ game: in every step, r vertices/edges appear at once, and Painter has to assign each of the r available colors to exactly one of these vertices.
• The case of edge colorings and r = 2 was previously studied by Marciniszyn, Mitsche, Stojaković (2005), who proved e.g. a threshold of n6/5 for the triangle (the threshold in the unbalanced game is n4/3).
![Page 32: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/32.jpg)
Balanced online vertex-coloring games
• Theorem (Prakash, S., 2007+)Let F be a clique or a cycle of arbitrary size.Then the threshold for the online balanced vertex-coloring game with respect to F and with r R 1 available colors is
• again r = 1 corresponds to Small Subgraphs theorem, and
• for r , the exponent tends to exponent for offline case.
• Key insight to go to arbitrary number of colors: • in every step, finding a valid coloring corresponds to finding a
perfect matching in some bipartite graph.
• use Hall‘s Theorem
![Page 33: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/33.jpg)
Two recent results…
• Theorem (Thomas, S., 2008+)[similar result for balanced edge-coloring game]
We can prove the upper bound for arbitrary r R 1 because no induction is needed.
• Theorem (Krivelevich, S., Steger, 2008+)The corresponding offline edge-coloring problem has the same threshold as the normal Ramsey problem [except …].
![Page 34: Online Ramsey Games in Random Graphs Reto Spöhel, ETH Zürich Joint work with Martin Marciniszyn and Angelika Steger TexPoint fonts used in EMF. Read the.](https://reader035.fdocuments.us/reader035/viewer/2022062719/56649ec05503460f94bcb4a7/html5/thumbnails/34.jpg)
Thank you! Questions?
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