A dominated pair condition for a digraph to be hamiltonian☆
Section snippets
Terminology and introduction
We shall assume that the reader is familiar with the standard terminology on digraphs and refer the reader to [2] for terminology not defined here. We only consider finite digraphs without loops and multiple arcs. Let be a digraph with vertex set and arc set . For an integer , will denote the set .
Let , be distinct vertices in . If there is an arc from to then we say that dominates and write . If dominates and dominates , then we write . If
Main result
The proof of Theorem 1.7 will be based on the following lemmas.
Lemma 2.1 Let be a path in and let be a path in . If can be multi-inserted into , then there is a -path in so that.[3]
Lemma 2.2 Let be a path in , and let . If cannot be inserted into , then .[3]
The proof of Theorem 1.7 Assume that is non-hamiltonian and is a longest cycle in . We first show that contains a -bypass. Suppose does not have one. Since is strong, must contain a cycle
Remark
To conclude the paper, we mention three related problems.
Remark 3.1 The remaining case of Conjecture 1.6 is .
Remark 3.2 Bang-Jensen, Guo and Yeo [1] proved that, if we replace the degree condition with in Conjecture 1.2, then is hamiltonian. So, it is natural to ask if there is an integer such that every strong digraph of order satisfying for every pair of dominated non-adjacent vertices , is hamiltonian.
It is easy to verify that if a strong digraph
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We thank the referees for their valuable comments and suggestions that improved the presentation considerably.
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