Abstract
The conservative number of a graph G is the minimum M such that G admits an orientation and a labeling of its edges with distinct numbers in {1,2,…,M} so that at each vertex of degree at least three, the sum of the labels of the incoming edges minus the sum of the labels of the outgoing edges is zero. A graph is conservative if its conservative number and its size are equal.In this work we determine the conservative number of several classes of trees and study the relation between conservative trees and graceful cycles with chords. We also show that for a given size of its base cycle, shell-type graphs with the maximum number of consecutive 4-faces are graceful.
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