On the existence of unparalleled even cycle systems

TL;DR

This paper establishes the existence conditions for unparalleled even cycle systems, showing they exist precisely when the order is divisible by twice the cycle length and greater than twice the cycle length.

Contribution

The paper provides a complete characterization of when unparalleled even cycle systems exist, filling a gap in combinatorial design theory.

Findings

Unparalleled 2t-cycle systems exist if and only if v > 2t > 2.

Such systems exist precisely when v is divisible by 2t.

The existence is characterized for all relevant parameters.

Abstract

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