On dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs☆
Introduction
This article is concerned with sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs. More specifically, we study several Meyniel-type criteria, that is, theorems asserting existence of hamiltonian cycles under certain conditions on the sums of degrees of non-adjacent vertices. There are numerous such criteria, and open problems, in general digraphs (see, e.g., [4], [5] and the references therein). Over the last few years, various analogues of these theorems and conjectures have been established for bipartite digraphs [1], [2], [3], [7], [8], [10], [11]. These results, generally speaking, do not follow from their non-bipartite analogues and require different arguments and techniques.
We begin with a short review of the relevant results to provide context for our present work. Throughout this paper, denotes a strongly connected balanced bipartite digraph of order (see Section 2 for details on notation and terminology).
The main feature of Meyniel-type criteria is that a degree condition be only imposed on pairs of non-adjacent vertices. The first such criterion in the bipartite setting was proved in [3].
Theorem 1.1 Let be as above, with , and suppose that for every pair of distinct vertices such that and . Then, is hamiltonian.
The lower bound of is sharp (see examples in [3]). The condition from Theorem 1.1 may be further strengthened, in the spirit of [5], by requiring that it be satisfied only by dominating and dominated pairs of vertices. This was done in [1].
Theorem 1.2 Let be as above, with , and suppose that whenever is a dominating or dominated pair. Then, is hamiltonian.
At this point, there are two natural questions: First, are the above assumptions enough to imply existence of cycles of all even lengths in , perhaps modulo some exceptional digraphs (Bondy’s metaconjecture)? And secondly, could we expect the same conclusion if the degree sum condition was only satisfied by the dominating pairs of vertices? The answer to the first question is positive. More precisely, we have the following result.
Theorem 1.3 Let be as above, with , and suppose that whenever is a dominating or dominated pair. Then, is either bipancyclic or a directed cycle of length .
The second question seems much harder. However, most recently, Wang and Wu [11] proposed an interesting variant of the degree sum condition that allows them to obtain hamiltonicity by only imposing the condition on dominating pairs of vertices.
Theorem 1.4 Let be a strongly connected balanced bipartite digraph of order , where , and let be an integer satisfying . Suppose that for every dominating pair of vertices in , Then, is hamiltonian.
The authors of [11] posed also several interesting problems related to the above theorems. Among them:
- (a)
Are the assumptions of Theorem 1.4 enough to imply bipancyclicity of ?
- (b)
Is there an integer such that is hamiltonian if the inequality is only imposed on the dominating pairs ?
The main goal of the present article is to prove the following positive answers to these two questions.
Theorem 1.5 If satisfies the hypotheses of Theorem 1.4, then is either bipancyclic or a directed cycle of length .
Theorem 1.6 Let be a strongly connected balanced bipartite digraph of order , where . Suppose that for every dominating pair of vertices in , Then, is hamiltonian.
Theorem 1.5, Theorem 1.6 are proved in Sections 4 Proof of, 3 Proof of, respectively. In the last section, we discuss some corollaries and open problems.
Section snippets
Notation and terminology
We consider digraphs in the sense of [4]: A digraph is a pair , where is a finite set (of vertices) and is a set of ordered pairs of distinct elements of , called arcs (i.e., has no loops or multiple arcs).
The number of vertices is the order of (also denoted by ). For vertices and from , we write to say that contains the ordered pair . If , then is called an in-neighbour of , and is an out-neighbour of . A pair of
Proof of Theorem 1.6
Throughout this section we assume that is a strongly connected balanced bipartite digraph with partite sets of cardinalities , which satisfies condition . The proof of Theorem 1.6 is based on the following four simple lemmas.
Lemma 3.1 Suppose that is non-hamiltonian. Then, for every vertex there exists a vertex such that is a dominating pair.
Proof For a proof by contradiction, suppose that contains a vertex which has no common out-neighbour with any other vertex in . We
Proof of Theorem 1.5
The proof of Theorem 1.5 is based on Theorem 1.3, Theorem 1.4, and the following result of Thomassen.
Theorem 4.1 Let be a strongly connected digraph of order , , such that whenever and are non-adjacent. Then, is either pancyclic, or a tournament, or is even and is isomorphic to .[9, Thm. 3.5]
Let be integers such that and . Note that then . Throughout this section, we assume that is a strongly connected balanced bipartite digraph of order , which
Final remarks
First of all, for the sake of completeness, let us note that analogues of Theorem 1.5, Theorem 1.6 for dominated pairs hold true as well. More precisely, we have the following results.
Theorem 5.1 Let be a strongly connected balanced bipartite digraph of order , where , and let be an integer satisfying . Suppose that for every dominated pair of vertices in , Then, is either bipancyclic or a directed cycle of length .
Theorem 5.2 Let be a
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.
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The research was partially supported by Natural Sciences and Engineering Research Council of Canada .