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

Abstract

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

7 pages. arXiv admin note: substantial text overlap with arXiv:1207.5643