A note on long non-hamiltonian cycles in one class of Digraphs

Let D be a strong digraph on n ≥ 4 vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) d(x) + d(y) ≥ 2n-1 and min{d
⁺
(x) + d
^-
(y), d
^-
(x)+d
⁺
(y)} ≥ n-1 for every pair of non-adjacent vertices x, y with a common in-neighbour or a common out-neighbour, then D is hamiltonian. In this note we show that: if D is not directed cycle and satisfies the condition (*), then D contains a cycle of length n - 1 or n - 2.