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A dominated pair condition for a digraph to be hamiltonian
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 n2 with the property that d(x)+d(y)2n1 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)n1 or d(x)n1, d(y)n, then D is hamiltonian. It is natural to ask if there is an integer k1 such that every strong digraph of order n satisfying either d(x)n+k, d(y)n1k, or d(x)n1k, 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 n5 and prove that every strong digraph with k=n4 satisfying the above condition is hamiltonian, except for one digraph on 5 vertices.

更新日期:2020-01-09

 

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