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A note on conservative galaxies, Skolem systems, cyclic cycle decompositions, and Heffter arrays
Goldfeder Ilan A
Tey Joaquin
Discrete Mathematics, 2018, 341(9): 2519-2528.
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 = vertical bar E(G)vertical bar. 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 equivalent to 0, 3 (mod 4), and M 1 otherwise. Consequently, given positive integers m(1), m(2),..., m(n) with m(i) >= 3 for 1
Skolem sequence; Conservative graph; Cyclic cycle system; Circulant graph; Sequenceable set; Heffter array
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