On magic distinct labellings of simple graphs

Abstract

A magic labelling of a graph G with magic sum s is a labelling of the edges of G by nonnegative integers such that for each vertex v∈V, the sum of labels of all edges incident to v is equal to the same number s. Stanley gave remarkable results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph G. The idea is illustrated in detail by dealing with three regular graphs and a non-regular graph. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct labellings. The idea can be carried out directly for graphs with no more than 9 edges.