Cluttered orderings for the complete bipartite graph

To minimize the access cost in large disk arrays (RAID) Cohen et al. [5–7] introduced (d, f)-cluttered orderings of various set systems, d, f ∈ N. In case of a graph this amounts to an ordering of the edge set such that the number of points contained in any d consecutive edges is bounded by the number f . For the complete graph, Cohen et al. gave some optimal solution for small parameters d [5] and introduced some general construction principle based on wrapped ∆-labellings [7]. In this paper, we investigate cluttered orderings for the complete bipartite graph. We adapt the concept of a wrapped ∆-labelling to the bipartite case and introduce the notion of a (d, f)-movement for subgraphs. From this we get a general existence theorem for cluttered orderings. The main result of this paper is the explicit construction of several infinite families of wrapped ∆-labellings leading to cluttered orderings for the corresponding bipartite graphs.