Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G   if its removal results in an acyclic subgraph. Let f(d,n)f(d,n) (fa(d,n)fa(d,n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d,n)K(d,n). This paper proves that for any integers d⩾2d⩾2 and n⩾1n⩾1f(d,n)=dforn=1,(ϕ⊙θ)(n)n+(ϕ⊙θ)(n-1)n-1for2⩽n⩽7,dnn+dn-1n-1+O(ndn-4)forn⩾8,fa(d,n)=f(d,n+1)forn⩾1,where (ϕ⊙θ)(n)=∑i|nϕ(i)θ(n/i)(ϕ⊙θ)(n)=∑i|nϕ(i)θ(n/i), i|ni|n means i divides n  , θ(i)=di+(-1)idθ(i)=di+(-1)id, ϕ(1)=1ϕ(1)=1 and ϕ(i)=i·∏j=1r(1-1/pj) for i⩾2i⩾2, where p1,…,prp1,…,pr are the distinct prime factors of i, not equal to 1.