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We discuss functions from edges and vertices of an undirected graph to anAbelian group. Such functions, when the sum of their values along any cycle iszero, are called balanced labelings. The set of balanced labelings forms anAbelian group. We study the structure of this group and the structure of twoclosely related to it groups: the subgroup of balanced labelings which consistsof functions vanishing on vertices and the corresponding factor-group. Thiswork is completely self-contained, except the algorithm for obtaining the3-edge-connected components of an undirected graph, for which we makeappropriate references to the literature.

Structure of the Group of Balanced Labelings on Graphs, its Subgroups and Quotient Groups