Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-01-08 , DOI: 10.1016/j.disc.2019.111794
Ruixia Wang; Jingfang Chang; Linxin Wu

In 1996, Bang-Jensen, Gutin, and Li proposed the following conjecture: If $D$ is a strong digraph of order $n$ where $n\ge 2$ with the property that $d\left(x\right)+d\left(y\right)\ge 2n-1$ for every pair of dominated non-adjacent vertices $\left\{x,y\right\}$, then $D$ is hamiltonian. In this paper, we give an infinite family of counterexamples to this conjecture. In the same paper, they showed that for the above $x,y$, if they satisfy the condition either $d\left(x\right)\ge n$, $d\left(y\right)\ge n-1$ or $d\left(x\right)\ge n-1$, $d\left(y\right)\ge n$, then $D$ is hamiltonian. It is natural to ask if there is an integer $k\ge 1$ such that every strong digraph of order $n$ satisfying either $d\left(x\right)\ge n+k$, $d\left(y\right)\ge n-1-k$, or $d\left(x\right)\ge n-1-k$, $d\left(y\right)\ge n+k$, for every pair of dominated non-adjacent vertices $\left\{x,y\right\}$, is hamiltonian. In this paper, we show that $k$ must be at most $n-5$ and prove that every strong digraph with $k=n-4$ satisfying the above condition is hamiltonian, except for one digraph on 5 vertices.

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