A Generalization of the Problem of Mariusz Meszka

Abstract

Mariusz Meszka has conjectured that given a prime $p=2n+1$ and a list $L$ containing $n$ positive integers not exceeding $n$ there exists a near $1$-factor in $K_p$ whose list of edge-lengths is $L$. In this paper we propose a generalization of this problem to the case in which $p$ is an odd integer not necessarily prime. In particular, we give a necessary condition for the existence of such a near $1$-factor for any odd integer $p$. We show that this condition is also sufficient for any list $L$ whose underlying set $S$ has size $1$, $2$, or $n$. Then we prove that the conjecture is true if $S=\{1,2,t\}$ for any positive integer $t$ not coprime with the order $p$ of the complete graph. Also, we give partial results when $t$ and $p$ are coprime. Finally, we present a complete solution for $t\le 11$.