  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 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-H$\ddot{\textrm{a}}$ggkvist Conjecture for oriented bipartite graphs
for some cases for which it is unsolved in general 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 small number of vertices.
