On dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs

Abstract

We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices.

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, $D$ denotes a strongly connected balanced bipartite digraph of order $2a$ (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 $D$ be as above, with $a\ge 2$, and suppose that

$$d\left(u\right)+d\left(v\right)\ge 3a$$

for every pair of distinct vertices $\{u,v\}$ such that $uv\notin A\left(D\right)$ and $vu\notin A\left(D\right)$. Then, $D$ is hamiltonian.

The lower bound of $3a$ 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 $D$ be as above, with $a\ge 3$, and suppose that

$$d\left(u\right)+d\left(v\right)\ge 3a$$

whenever $\{u,v\}$ is a dominating or dominated pair. Then, $D$ 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 $D$, 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 $D$ be as above, with $a\ge 3$, and suppose that

$$d\left(u\right)+d\left(v\right)\ge 3a$$

whenever $\{u,v\}$ is a dominating or dominated pair. Then, $D$ is either bipancyclic or a directed cycle of length $2a$.

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 $D$ be a strongly connected balanced bipartite digraph of order $2a$, where $a\ge 3$, and let $k$ be an integer satisfying $max\{1,\frac{a}{4}\}<k\le \frac{a}{2}$. Suppose that for every dominating pair $\{u,v\}$ of vertices in $D$,

$$d\left(u\right)\ge 2a-k\mathrm{and}d\left(v\right)\ge a+k,\mathrm{or}d\left(u\right)\ge a+k\mathrm{and}d\left(v\right)\ge 2a-k.$$

Then, $D$ is hamiltonian.

The authors of [11] posed also several interesting problems related to the above theorems. Among them:

• Are the assumptions of Theorem 1.4 enough to imply bipancyclicity of $D$?

• Is there an integer $k\ge 0$ such that $D$ is hamiltonian if the inequality $d\left(u\right)+d\left(v\right)\ge 3a+k$ is only imposed on the dominating pairs $\{u,v\}$?

The main goal of the present article is to prove the following positive answers to these two questions.

Theorem 1.5

If $D$ satisfies the hypotheses of Theorem 1.4, then $D$ is either bipancyclic or a directed cycle of length $2a$.

Theorem 1.6

Let $D$ be a strongly connected balanced bipartite digraph of order $2a$, where $a\ge 2$. Suppose that for every dominating pair $\{u,v\}$ of vertices in $D$,

$$d\left(u\right)+d\left(v\right)\ge 3a+1.$$

Then, $D$ 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.