Oriented Bipartite Graphs and the Goldbach Graph

In this paper we study oriented bipartite graphs. In particular, we introduce bitransitive graphs and bitournaments. Several characterizations of bitransitive bitournaments are obtained. Next we prove the Caccetta-H$\ddot{\textrm{a}}$ggkvist Conjecture for oriented bipartite graphs for some cases for which it is unsolved in general. We introduce the concept of oriented odd-even graphs and (undirected) 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. Next we introduce Goldbach graphs which are special type of odd-even graphs. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other related conjectures are related to various parameters of Goldbach graphs. We study nature of degrees of vertices and independent sets of Goldbach graphs. Finally we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for small number of vertices.