Oriented Bipartite Graphs and the Goldbach Graph

Abstract

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-Häggkvist 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 and characterize (oriented) bipartite graphs in terms of them.In fact, we show that any (oriented) bipartite graph can be represented by some (oriented) odd-even graph. We obtain some conditions for connectedness of odd-even graphs. Finally 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.

The Structure of 2-Colored Best Match Graphs

“…Several papers give sufficient conditions for bipartite digraphs, in terms of the number of edges, to have cycles and paths with specified properties. These conditions can be viewed as digraph versions or variants of similar conditions on undirected bipartite graphs which were widely studied since the 1980's; see [1,2,3,4,6,5,12,16,17,18,21,22,23]. We recall those which are related to the present investigation.…”

“…A vertex u is minimal if there is no vertex v such that v < u and vu ∈ E, that is, if N − (u) = ∅; maximality for vertices is defined analogously. A sufficient condition for an oriented digraph to have a topological ordering is to be acyclic, that is, there is no directed cycle in the digraph; see [6]. Acyclic oriented digraphs are odd-even graphs; see [6,Theorem 3.4] and [13].…”

“…A sufficient condition for an oriented digraph to have a topological ordering is to be acyclic, that is, there is no directed cycle in the digraph; see [6]. Acyclic oriented digraphs are odd-even graphs; see [6,Theorem 3.4] and [13]. For a pair (A, O) where A is a finite set of non-negative even integers and O is a set a positive odd integers, the associated odd-even…”

“…We will also use the term "almost 2-cBMG" when each but at most one vertex has an out-neighbor. [1,8], [1,9], [1,10], [2,8], [3,9], [4,10], [5,9], [6,9], [5,10], [6,10], [7,1], [7,2], [7,3], [7,4], [7,5], [7,6], [8,2], [9,3], [10,4] >.…”

“…In the study of best match graphs, the It should be noticed however that N1 and N3 had not been considered in the literature on graph theory until the discovery of their links to evolutionary relatedness via phylogenetic trees. N2 was introduced, but only marginally studied, under the name "bi-transitive" property in the preprint [6].…”