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The acyclic orientation game on random graphs

Summary: The acyclic orientation game
on random graphs
Noga Alon
Zsolt Tuza
Dedicated to Professor Paul Erdos on the occasion of his 80th
It is shown that in the random graph Gn,p with (fixed) edge probability
p > 0, the number of edges that have to be examined in order to identify an
acyclic orientation is (n log n) almost surely. For unrestricted p, an upper
bound of O(n log3
n) is established. Graphs G = (V, E) in which all edges
have to be examined are considered, as well.
1 Introduction
In this note we investigate the typical length of the following 2-person game. Given
a graph G = (V, E), in each step of the game player A (Algy) selects an edge e E
and player S (Strategist) orients e in the way he likes; the only restriction is that
S must not create a directed circuit. The game is over when the actually obtained
partial orientation of G extends to a unique acyclic orientation. The goal of A is
to locate such an orientation with as few questions as possible, while S aims at the


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics