Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs

Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced matching number and the ordered matching number of G, respectively. We characterize all bipartite graphs G with indm(G) = ordm(G). We establish the Castelnuovo-Mumford regularity of powers of edge ideals and depth of powers of cover ideals for such graphs. We also give formulas for the count of connected non-isomorphic spanning subgraphs of the complete bipartite graph Km,n for which indm(G) = ordm(G) = 2, with an explicit expression for the count when m = 2,3,4 and m <= n.