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

Abstract

The conservative number of a graph G is the minimum positive integer M, such that G admits an orientation and a labeling of its edges by distinct integers in {1,2,…,M}, such that at each vertex of degree at least three, the sum of the labels on the in-coming edges is equal to the sum of the labels on the out-going edges. A graph is conservative if M=|E(G)|. It is worth noting that determining whether certain biregular graphs are conservative is equivalent to find integer 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.