Representations of graphs modulo <i>n</i>

Abstract

Abstract A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n , the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.