| author: | Martin Marciniszyn, Dieter Mitsche and Miloš Stojaković |
|---|---|
| title: | Balanced Avoidance Games on Random Graphs |
| keywords: | |
| abstract: |
We introduce and study balanced online graph avoidance
games on the random graph process. The game is played by a
player we call Painter. Edges of the complete graph with
n
vertices are revealed two at a time in a random
order. In each move, Painter immediately and irrevocably
decides on a balanced coloring of the new edge pair: either
the first edge is colored red and the second one blue or
vice versa. His goal is to avoid a monochromatic copy of a
given fixed graph
H
in both colors for as long as possible. The game ends
as soon as the first monochromatic copy of
H
has appeared. We show that the duration of the game
is determined by a threshold function
m
. More precisely, Painter will asymptotically almost
surely (a.a.s.) lose the game after
H
= m
H
(n)
m = ω(m
edge pairs in the process. On the other hand, there
is an essentially optimal strategy, that is, if the game
lasts for
H
)
m = o(m
moves, then Painter will a.a.s. successfully
avoid monochromatic copies of
H
)
H
using this strategy. Our attempt is to determine the
threshold function for certain graph-theoretic structures,
e.g., cycles.
|
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| reference: | Martin Marciniszyn and Dieter Mitsche and Miloš Stojaković (2005), Balanced Avoidance Games on Random Graphs, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 329-334 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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