A note on conservative galaxies,Skolem systems,cyclic cycle decompositions,and Heffter arrays |
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Authors: | Ilan A Goldfeder Joaquín Tey |
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Institution: | Universidad Autónoma Metropolitana, Iztapalapa Av. San Rafael Atlixco 186, col. Vicentina, 09340 Iztapalapa, Mexico |
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Abstract: | The conservative number of a graph is the minimum positive integer , such that admits an orientation and a labeling of its edges by distinct integers in , 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 . 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 is for , , and otherwise. Consequently, given positive integers , , …, with for , we construct a cyclic -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 -cycle system of the complete graph , where . Also, we prove necessary and sufficient conditions for the existence of a cyclic -cycle system of , where is a 1-factor. Furthermore, we give a sufficient condition for a subset of to be sequenceable. |
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Keywords: | Skolem sequence Conservative graph Cyclic cycle system Circulant graph Sequenceable set Heffter array |
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