The existence of 2-balanced Mendelsohn triple systems

摘要

A Mendelsohn triple system MTS(v, b) is a collection of b cyclic triples (blocks) on a set of v points. It is j-balanced for j = 1, 2, 3 when any two points, ordered pairs, or cyclic triples (resp.) are contained in the same or almost the same number of blocks (difference at most one). A (2, 3)-balanced Mendelsohn triple system is an MTS(v, b) that is both 2-balanced and 3-balanced. Employing large sets of Mendelsohn triple systems and partitionable Mendelsohn candelabra systems, we completely determine the spectrum for which a 2-balanced Mendelsohn triple system exists. Meanwhile, we determine the existence spectrum for a (2, 3)-balanced Mendelsohn triple system. (C) 2019 Elsevier B.V. All rights reserved.