Introduction of Bipartite Graph Definition and Properties A bipartite graph is a type of graph in which the set of vertices can be divided into two distinct sets such that no two vertices within the same set are adjacent. In other words, every edge in the graph connects a vertex from one set to a vertex in the other set. A bipartite graph G = ( V , E ) G = (V, E) G = ( V , E ) consists of: Two disjoint sets of vertices, say U U U and V V V , such that U ∩ V = ∅ U \cap V = \emptyset U ∩ V = ∅ . Edges that only connect vertices between U U U and V V V , meaning no edge exists within U U U or within V V V . A bipartite graph can be represented as: G = ( U , V , E ) G = (U, V, E) G = ( U , V , E ) where: U U U and V V V are the two sets of vertices. E E E is the set of edges where each edge ( u , v ) (u, v) ( u , v ) satisfies u ∈ U u \in U u ∈ U and v ∈ V v \in V v ∈ V . Conditions for Bipartiteness A graph is bipartite if and only if it does not contain an odd-length cycle . T...