Given a bipartite graph G = (V, W, E), a 2-layered drawing consists of placing nodes in the first node set V on a straight line L-1 and placing nodes in the second node set Won a parallel line L-2. For a given ordering of nodes in Won L2, the one-sided crossing minimization problem asks to find an ordering of nodes in V on L-1 so that the number of arc crossings is minimized. A well-known lower bound LB on the minimum number of crossings is obtained by summing up min{c(uv), c(vu)} over all node pairs u, v is an element of V, where c(uv) denotes the number of crossings generated by arcs incident to u and v when u precedes v in an ordering. In this paper, we prove that there always exists a solution whose crossing number is at most (1.2964 + 12/(delta - 4))LB if the minimum degree delta of a node in V is at least 5. (c) 2004 Elsevier B.V. All rights reserved.