Online Vertex Colorings of Random Graphs Without Monochromatic Subgraphs Reto Spöhel, ETH Zurich...
-
date post
21-Dec-2015 -
Category
Documents
-
view
213 -
download
0
Transcript of Online Vertex Colorings of Random Graphs Without Monochromatic Subgraphs Reto Spöhel, ETH Zurich...
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 *