Executive Summary : | This project aims to study bipartite graphs and their applications, as well as the structural properties of oriented bitransitive graphs and oriented bipartite graphs. The researchers have already obtained some structural properties for oriented bitransitive graphs and oriented bipartite graphs, which will help solve Seymour's 2nd neighborhood conjecture for oriented bipartite graphs. They also discovered the Hamiltonian property for some bipartite graphs closely related to Goldbach graphs, leading to a sequence of even natural numbers up to 1000. The project also focuses on the study of interval catch digraphs (ICD), introduced by Maehara in 1984. They have recently studied three subclasses of ICD: central ICD, oriented ICD, and proper ICD. They disproved a Maehara conjecture on the characterization of central ICD and obtained the characterization theorem. In addition to these studies, the researchers are also interested in studying bipartite interval catch digraphs and their subclasses. The researchers have already proven the existence of a sequence of even natural numbers up to 1000, where one is the sum of two primes and the other is the difference of these primes. Overall, this project aims to continue the research on bipartite graphs, Goldbach graphs, and interval catch digraphs, with a focus on their applications and structural properties. |