Vertex-magic labeling of trees and forests

• Published in: Discrete mathematics Vol. 261; no. 1; pp. 285 - 298
• Main Authors: Gray, I.D., MacDougall, J., McSorley, J.P., Wallis, W.D.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 28-01-2003
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/S0012-365X(02)00475-2

A
vertex-magic total labeling of a graph
G(
V,
E) is a one-to-one map
λ from
E∪
V onto the integers {1,2,…,|
E|+|
V|} such that
λ(x)+∑λ(xy),
where the sum is over all vertices
y adjacent to
x, is a constant, independent of the choice of vertex
x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite different from the conjectured behavior of
edge-magic total labelings of these graphs. We pay special attention to the case of so-called
galaxies, forests in which every component tree is a star.