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 is a strong digraph of order where with the property that for every pair of dominated non-adjacent vertices , then 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 , if they satisfy the condition either , or , , then is hamiltonian. It is natural to ask if there is an integer such that every strong digraph of order satisfying either , , or , , for every pair of dominated non-adjacent vertices , is hamiltonian. In this paper, we show that must be at most and prove that every strong digraph with satisfying the above condition is hamiltonian, except for one digraph on 5 vertices.