Oriented Bipartite Graphs and the Goldbach Graph

TL;DR

This paper explores properties of oriented bipartite graphs, introduces new graph classes like bitransitive and odd-even graphs, and connects these to number theory conjectures such as Goldbach's, providing new characterizations and solving specific problems.

Contribution

It introduces and characterizes new classes of bipartite graphs, relates Goldbach graphs to number theory conjectures, and solves the graph-isomorphism problem for acyclic bitournaments.

Findings

Characterization of bitransitive bitournaments as acyclic

Linear-time algorithm for graph-isomorphism of acyclic bitournaments

Goldbach's conjecture equivalent to Goldbach graphs' connectedness

Abstract

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.