A Generalization of the Problem of Mariusz Meszka

作者：Pasotti Anita; Pellegrini Marco Antonio

来源：Graphs and Combinatorics, 2016, 32(1): 333-350.

DOI：10.1007/s00373-015-1563-0

• Near 1-factor
• Complete graph
• Edge-length
• Cyclic decomposition
• Skolem sequence

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

• 出版日期2016-1