A Generalization of the Problem of Mariusz Meszka

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 ≤ 11.