摘要

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 <= i <= n, we construct a cyclic (m(1), m(2), ..., m(n))-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 (m(1), m(2),..., m(n))-cycle system of the complete graph K2M+1, where M = Sigma(n)(i=1) m(i). Also, we prove necessary and sufficient conditions for the existence of a cyclic (m(1), m(2), m(n))-cycle system of K2M+2 - F, where F is a 1-factor. Furthermore, we give a sufficient condition for a subset of Z(v) \ {0} to be sequenceable.

  • 出版日期2018-9

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