 
Summary: Representation Numbers of Sparse Graphs
Reza Akhtar
Dept. of Mathematics
Miami University, Oxford, OH 45056, USA
reza@calico.mth.muohio.edu
September 24, 2010
Abstract
We study the representation number for various sparse graphs; in particular,
we give an exact formula for graphs with a single edge and complete binary trees
and an improved lower bound for the representation number of the hypercube.
We also study the prime factorization of the representation number of graphs
with a single edge.
1 Introduction
A finite graph G is said to be representable modulo r if there exists an injective map
f : V (G) {0, 1, . . . , r  1} such that for all u, v V (G), gcd(f(u)  f(v), r) = 1
or equivalently, if there exists an injective map f : V (G) Zr such that for all
u, v V (G), f(u)  f(v) is a unit of (the ring) Zr. The representation number of G,
denoted rep(G), is the smallest positive integer r modulo which G is representable. If
we define the unitary Cayley graph Cay(n) on n vertices to be the graph with vertex
set {0, 1, . . . , n  1}, two of whose vertices are adjacent if and only if their difference
