The seating couple problem in even case

TL;DR

This paper investigates the seating couple problem for an even number of seats, providing necessary conditions, constructive solutions for specific cases, and proposing conjectures and open problems in the graph theory context.

Contribution

It offers the first complete constructive solutions for certain list configurations and introduces necessary conditions for the existence of perfect matchings with specified edge-lengths.

Findings

Necessary conditions for solution existence

Constructive solutions for lists with one or two distinct elements

Solutions for lists of consecutive integers with equal multiplicities

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.