A note on conservative galaxies, Skolem systems, cyclic cycle decompositions, and Heffter arrays

In this work we show that the conservative number of a galaxy (a disjoint union of stars) of size M is M for M≡0, 3(mod4), and M+1 otherwise. Consequently, given positive integers m1, m2, …, mn with mi≥3 for 1≤i≤n, we construct a cyclic (m1,m2,…,mn)-cycle system of infinitely many circulant graphs, generalizing a result of Bryant, Gavlas and Ling (2003). In particular, it allows us to construct a cyclic (m1,m2,…,mn)-cycle system of the complete graph K2M+1, where M=∑i=1nmi. Also, we prove necessary and sufficient conditions for the existence of a cyclic (m1,m2,…,mn)-cycle system of K2M+2−F, where F is a 1-factor. Furthermore, we give a sufficient condition for a subset of Zv∖{0} to be sequenceable.