A counterexample to a generalization of Richardson's theorem (Q1088675)

The following result has apparently been obtained by \textit{M. Kwasnik} in Discuss. Math. 4, 11-13 (1981; Zbl 0509.05048): If a finite digraph has no circuits of length \(nk+r\) for \(n=0,1,2,..\). and \(0<r<k\), then it has a k-kernel (k\(\geq 2\); Richardson's theorem refers to the case \(k=2).\) The present note shows that the result is not always valid for \(k>2\) unless an additional condition is imposed, such that the digraph be strongly connected [loc. cit.].