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The Representation Number of the Random Graph Reza Akhtar

Summary: The Representation Number of the Random Graph
Reza Akhtar
Dept. of Mathematics
Miami University
Oxford, OH 45056, USA
Je#rey R. Cooper
Dept. of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
851 S. Morgan Street, Chicago, IL 606077045
May 25, 2011
A graph G is said to have a representation modulo r if there exists an
injective map f : V (G) # {0, . . . , r - 1} such that vertices u, v # V (G) are
adjacent if and only if gcd(f(i) - f(j), r) = 1. Its representation number,
denoted rep(G), is the smallest r modulo which it has such a representation.
We prove that ln rep(G(n, 1/2)) = #(n), where G(n, 1/2) is the random graph
on n vertices with edge probability 1/2. As part of the proof, we show that the
product dimension of G(n, 1/2) is #(n ln n).


Source: Akhtar, Reza - Department of Mathematics and Statistics, Miami University (Ohio)


Collections: Mathematics