On longest non-Hamiltonian cycles in digraphs with the conditions of Bang-Jensen, Gutin and Li

Abstract

Let $D$ be a strongly connected directed graph of order $n\ge 4$. In Bang-Jensen et al. (1996), (J. of Graph Theory 22 (2) (1996) 181–187), J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If $(\ast )$$d\left(x\right)+d\left(y\right)\ge 2n-1$ and $min\{d\left(x\right),d\left(y\right)\}\ge n-1$ for every pair of non-adjacent vertices $x,y$ with a common in-neighbour or $(\ast \ast )min\{d^{+}\left(x\right)+d^{-}\left(y\right),d^{-}\left(x\right)+d^{+}\left(y\right)\}\ge n$ 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 paper we show that: (i) if $D$ satisfies condition $(\ast )$ and the minimum semi-degree of $D$ at least two or (ii) if $D$ is not directed cycle and satisfies condition $(\ast \ast )$, then either $D$ contains a cycle of length $n-1$ or $n$ is even and $D$ is isomorphic to the complete bipartite digraph or to the complete bipartite digraph minus one arc.