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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
Jeffrey R. Cooper
Dept. of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
851 S. Morgan Street, Chicago, IL 60607-7045
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