The seating couple problem in even case

Abstract: In this paper we consider the seating couple problem with an even number of seats, which, using graph theory terminology, can be stated as follows. Given a positive even integer $v=2n$ and a list $L$ containing $n$ positive integers not exceeding $n$, is it always possible to find a perfect matching of $K_v$ whose list of edge-lengths is $L$? Up to now a (non-constructive) solution is known only when all the edge-lengths are coprime with $v$. In this paper we firstly present some necessary conditions for the existence of a solution. Then, we give a complete constructive solution when the list consists of one or two distinct elements, and when the list consists of consecutive integers $1,2,\ldots,x$, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.