The algebra of basic covers of a graph G, denoted by \A(G), was introduced by
Juergen Herzog as a suitable quotient of the vertex cover algebra. In this
paper we show that if the graph is bipartite then \A(G) is a homogeneous
algebra with straightening laws and thus is Koszul. Furthermore, we compute the
Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence
we get new upper bounds on the arithmetical rank of monomial ideals of pure
codimension 2. Finally, we characterize the Cohen-Macaulay property and the
Castelnuovo-Mumford regularity of the edge ideal of a certain class of graphs.