A dominated pair condition for a digraph to be hamiltonian

https://doi.org/10.1016/j.disc.2019.111794Get rights and content

Abstract

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.

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 D be a digraph with vertex set V(D) and arc set A(D). For an integer n, [n] will denote the set {1,2,,n}.

Let x, y be distinct vertices in D. If there is an arc from x to y then we say that x dominates y and write xy. If x dominates y and y dominates x, then we write xy. If x

Main result

The proof of Theorem 1.7 will be based on the following lemmas.

Lemma 2.1

[3]

Let P be a path in D and let Q=v1v2vt be a path in DV(P). IfP can be multi-inserted into Q, then there is a (v1,vt)-pathR inD so thatV(R)=V(P)V(Q).

Lemma 2.2

[3]

Let Q=v1v2vt be a path in D, and let wV(D)V(Q). Ifw cannot be inserted into Q, then dQ(w)t+1.

The proof of Theorem 1.7

Assume that D is non-hamiltonian and C=x1x2xmx1 is a longest cycle in D. We first show that D contains a C-bypass. Suppose D does not have one. Since D is strong, D must contain a cycle Z

Remark

To conclude the paper, we mention three related problems.

Remark 3.1

The remaining case of Conjecture 1.6 is 1kn5.

Remark 3.2

Bang-Jensen, Guo and Yeo [1] proved that, if we replace the degree condition d(x)+d(y)2n1 with d(x)+d(y)52n4 in Conjecture 1.2, then D is hamiltonian. So, it is natural to ask if there is an integer k1 such that every strong digraph of order n satisfying d(x)+d(y)2n1+k for every pair of dominated non-adjacent vertices {x,y}, 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.

References (6)

There are more references available in the full text version of this article.

This work is supported by the National Natural Science Foundation for Young Scientists of China (11401354).

View full text