A Note on Long non-Hamiltonian Cycles in One Class of Digraphs

Abstract: Let $D$ be a strong digraph on $n\geq 4$ vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if () $d(x)+d(y)\geq 2n-1$ and $$min {d^+(x)+ d^{-(y),d^-(x)+} d^+(y)}\geq n-1$$ 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 $n-2$.