On cycles in the sequence of unitary Cayley graphs

Abstract

For n∈ N let p k ( n) be the number of induced k-cycles in the Cayley graph Cay ( Z n , U n ), where Z n is the ring of integers mod n and U n=Z n ∗ is the group of units mod n . Our main result is: Given r∈ N there is a number m( r), depending only on r, with r ln r⩽m(r)⩽9r! such that p k ( n)=0 if k⩾ m( r) and n has at most r prime divisors. As a corollary we deduce the existence of non-trivial arithmetic functions f with the properties: f is a Z- linear combination of multiplicative arithmetic functions. f( n)=0 for every n with at most r different prime divisors. We also prove the chromatic uniqueness of Cay ( Z n , U n ) for n a prime power.