Given integers , the *2-stage Shuffle-Exchange graph/network*, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).

Given integers , the *-stage Shuffle-Exchange graph/network*, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).

Let be the smallest integer such that the graph is rearrangeable.

**Problem**Find .

**Conjecture**.

A *mask* for the graph is a -regular bipartite multigraph with the bipartition . The graph is said to be *rearrangeable* if for every its mask there exists a collection, called *routing*, of corresponding mutually edge-disjoint paths in connecting its end parts. (For simplicity, we do not provide here a more general definition for rearrangeability of graphs.)

Note that is a simple -partite graph with vertices and edges, and any route for it consists exactly of paths. Also, is equivalent to rearrangeability of .

**Figure 1.** Examples of multistage Shuffle-Exchange graphs.

For example, according to the conjecture, the graph (see Fig. 1) is rearrangeable, which is a well known result.

The problem and conjecture are equivalent "graph-theoretic" forms of remarkable Shuffle-Exchange (SE) problem and conjecture due to the following identity (that is not hard to show by normal reasoning):

**Theorem**.

The definition of and more on SE problem/conjecture including the other 2 main forms of them, combinatorial and group-theoretic, and a survey of results can be found here.

## Bibliography

*[S71] H.S. Stone, *Parallel processing with the perfect shuffle*, IEEE Trans. on Computers **C-20** (1971), 153-161.

*[B75] V.E. Beneš, *Proving the rearrangeability of connecting networks by group calculation*, Bell Syst. Tech. J. **54** (1975), 421-434.

* indicates original appearance(s) of problem.