On the existence of unparalleled even cycle systems

Abstract

A 2t-cycle system of order v is a set C of cycles whose edges partition the edge-set of Kv − I (i.e., the complete graph minus the 1-factor I). If v ≡ 0 (mod 2t), a set of v/2t vertex-disjoint cycles of C is a parallel class. If C has no parallel classes, we call such a system unparalleled.We show that there exists an unparalleled 2t-cycle system of order v ≡ 0 (mod 2t) if and only if v > 2t > 2.

On the minisymposium problem

“…Theorem 3.1 is more than just an existence result; a recursive procedure can be extracted from the proof to algorithmically build the edge decompositions. We have a more direct construction of all HMSF(v, m) which uses difference methods and decompositions of Cayley graphs [16].…”