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A dominated pair condition for a digraph to be hamiltonian
Discrete Mathematics ( IF 0.728 ) 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≥2 with the property that d(x)+d(y)≥2n−1 for every pair of dominated non-adjacent vertices {x,y}, 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(x)≥n, d(y)≥n−1 or d(x)≥n−1, d(y)≥n, then D is hamiltonian. It is natural to ask if there is an integer k≥1 such that every strong digraph of order n satisfying either d(x)≥n+k, d(y)≥n−1−k, or d(x)≥n−1−k, d(y)≥n+k, for every pair of dominated non-adjacent vertices {x,y}, 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.
更新日期:2020-01-09

 

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