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Vertex-Magic Edge Labeling

A vertex-magic edge labeling of a graph G=(V,E) is a bijection

f:E->{1,2,...,|E|}

for which the vertex weight

w(v)=sum_(e in E e incident to v)f(e)

has the same value for every graph vertex v. A graph admitting such a labeling is a magic graph in a common graph-labeling sense.

The word "vertex-magic" refers to the constant vertex weights, while "edge labeling" specifies where the labels are placed. This construction is distinct from an edge-magic total labeling, which labels both graph vertices and graph edges and makes the edge weights constant.

See also

Edge-Magic Total Labeling, Magic Graph, Magic Labeling

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References

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin., Dynamic Survey DS6, Oct. 30, 2025. https://doi.org/10.37236/27.

Cite this as:

Weisstein, Eric W. "Vertex-Magic Edge Labeling." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Vertex-MagicEdgeLabeling.html

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