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The design of large disk array architectures leads to interesting combinatorial problems. Minimizing the number of disk operations when writing to consecutive disks leads to the concept of “cluttered orderings” which were introduced for the complete graph by Cohen
*et al*. (2001). Mueller
*et al*. (2005) adapted the concept of wrapped Δ-labellings to the complete bipartite case. In this paper, we give some sequence in order to generate wrapped Δ-labellings as cluttered orderings for the complete bipartite graph. New sequence we give is different from the sequences Mueller
*et al*. gave, though the same graphs in which these sequences are labeled.

The desire to speed up secondary storage systems has lead to the development of disk arrays which achieve performance through disk parallelism. While performance improves with increasing numbers of disks, the chance of data loss coming from catastrophic failures, such as head crashes and failures of the disk controller electronics, also increases. To avoid high rates of data loss in large disk arrays, one includes redundant information stored on additional disks―also called check disks―which allows the reconstruction of the original data― stored on the so-called information disks―even in the presence of disk failures. These disk array architectures are known as redundant arrays of independent disks (RAID) (see [

Optimal erasure-correcting codes using combinatorial framework in disk arrays are discussed in [

As

In this paper, we make label to the vertex of a bipartite graph. For example, we make label 1, 3, 0 and −1, respectively, to four vertices a, b, c and d of a bipartite graph in

In a RAID system disk writes are expensive operations and should therefore be minimized. In many applications there are writes on a small fraction of consecutive disks―say d disks―where d is small in comparison to k, the number of information disks. Therefore, to minimize the number of operations when writing to d consecutive information disks one has to minimize the number of check disks―say f―associated to the d information disks.

Let

Let

In the following,

two subsets denoted by V and W. Any edge of the edge set E contains exactly one point of V and W respectively. Let

Here,

Proposition 1. ([

For example,

Definition 1. Let G be a graph with edge set

In order to assemble such (d, f)-movements of certain subgraphs to a (d, f)-cluttered ordering, we need some notion of consistency. Let

Now, for each

also denote

by specifying its edge set

Definition 2. With above notation, a (d, f)-movement of

According to Definition 1, such a (d, f)-movement is given by some permutation

defines a bijection

Then

Having such a consistent

Proposition 2. ([

In this section, we define an infinite family of bipartite graphs which allow (d, f)-movements with small f. In order to ensure that these (d, f)-movements are consistent with some translation parameter

Let h and t be two positive integers. For each parameter f and t, we define a bipartite graph denoted by

The edge set E is partitioned into subsets

The t subgraphs defined by the edge sets E_{s},

Proposition 3. ([

By Proposition 1 a Δ-labelling of the graph

Definition 3. Let

as multisets in

For the graphs

hold for

Proposition 4. ([

In this section, we construct some infinite families of such wrapped Δ-labellings. By applying Proposition 2 we get explicite (d, f)-cluttered orderings of the corresponding bipartite graphs. For these results in this section, we refer to [

We define a wrapped Δ-labelling of

and 3t edges. For a fixed t, we define

where the integers in the first components are considered modulo 3t. We now compute the difference list

Obviously, the wrapped-condition (7) relative to

Theorem 5. ([

Theorem 6. ([

We define a wrapped Δ-labelling of

and, on the vertices

where we set

All integers are considered modulo 10t. Note that

Theorem 7. ([

Theorem 8. ([

We define in this section a wrapped Δ-labelling for

4h vertices and

by specifying the first component of Δ on the vertices

and on the vertices

where we set

All integers are considered modulo

Theorem 9. ([

In this section, we define a wrapped Δ-labelling of

and, on the vertices

where we set

All integers are considered modulo 21t. Note that

We now compute the differences of Δ using the notation from (1):

We now compute the difference list

From this one easily checks that the twenty-two lists cover all numbers in

Theorem 10. Let t be a positive integer. For all t there is a (d, f)-cluttered ordering of the complete bipartite graph

Using the same edge ordering of

Theorem 11. Let t be a positive integer. For all t there is a (d, f)-cluttered ordering of the complete bipartite graph

For example, we get a (21, 12)-cluttered ordering of

In conclusion, we give a new sequence for construction of wrapped Δ-labellings.

We thank the Editor and the referee for their comments.