On cycles in the sequence of unitary Cayley graphs

For n∈N let pk(n) be the number of induced k-cycles in the Cayley graph Cay (Zn,Un), where Zn is the ring of integers mod n and Un=Zn∗ 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 pk(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 (Zn,Un) for n a prime power.