In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.

在本文中，我们研究了有向二部图。特别是，我们引入了“双传递”图。获得了双传递bitournaments的几个特征。我们证明了双传递的 bitournaments 等价于非循环的 bitournaments。作为应用，我们用哈密顿路径来表征非循环 bitournaments，确定给定阶次的非同构非循环 bitournaments 的数量，并在线性时间内解决非循环 bitournaments 的图同构问题。接下来，我们证明了著名的有向二部图的 Caccetta-Häggkvist 猜想，在某些情况下，一般来说，对于有向图，它是未解决的。我们还介绍了无向和有向“奇偶”图的概念。我们根据它们来表征二部图和无环定向二部图。实际上，我们证明了任何二部图（无环定向二部图）都可以用一些奇偶图（面向奇偶图）来表示。我们得到了奇偶图连通性的一些条件。这项对奇偶图及其连通性的研究是由我们称为“哥德巴赫图”的特殊奇偶图系列所激发的。我们证明著名的哥德巴赫猜想等价于哥德巴赫图的连通性。其他几个数论猜想（例如孪生素数猜想）与哥德巴赫图的各种参数有关，促使我们研究这些图的顶点度和独立集的性质。最后，我们观察了一些与哥德巴赫图相关的奇偶图的哈密顿性质，这些奇偶图的顶点数量很少。