Question: does a bipartite graph have to be connected

Answer: No, One often writes G = ( U , V , E ) (\displaystyle G=(U,V,E)) to denote a bipartite graph whose partition has the parts U (\displaystyle U) and V (\displaystyle V) , with E (\displaystyle E) denoting the edges of the graph. If a bipartite graph is not connected, it may have more than one bipartition; in this case, the ( U , V , E ) (\displaystyle (U,V,E)) notation is helpful in specifying one particular bipartition that may be of importance in an application. If U = V (\displaystyle U = V ) , that is, if the two subsets have equal cardinality, then G (\displaystyle G) is called a balanced bipartite graph. If all vertices on the same side of the bipartition have the same degree, then G (\displaystyle G) is called biregular.