In this paper we consider the seating couples 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 Kv 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,…,x, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.
The seating couples problem in the even case
Pasotti A.;Pellegrini M. A.
2024-01-01
Abstract
In this paper we consider the seating couples 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 Kv 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,…,x, each one appearing with the same multiplicity. Finally, we propose a conjecture and some open problems.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
