A conservative graph is a finite undirected graph whose 
edges admit an orientation
and a bijective labeling by the integers
1, 2, ..., 
such that the sum of labels on edges directed into
each vertex equals the sum on edges
directed out of that vertex. Such an oriented edge
labeling is called a conservative labeling
(Bange et al. 1980; Gardner 2023, p. 1).
The dodecahedral graph is conservative and has a labeling using the integers 1 through 30 (Gardner
2023, pp. 2-3).
See also
Graph Orientation
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References
Bange, D. W.; Barkauskas, A. E.; and Slater, P. J. "Conservative Graphs." J. Graph Th. 4, 81-91, 1980. https://doi.org/10.1002/jgt.3190040109.Gardner,
R. "Section 6.2. Conservative Graphs." Introduction to Graph Theory--Class
Notes. Rev. Jan. 15, 2023. https://faculty.etsu.edu/gardnerr/5347/Notes/Pearls-GT-6-2.pdf.
Cite this as:
Weisstein, Eric W. "Conservative Graph."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConservativeGraph.html
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