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

TL;DR

This paper investigates the algebraic structure of balanced labelings on graphs, focusing on their groups, subgroups, and quotient groups, and provides a comprehensive analysis of their properties.

Contribution

It offers a detailed study of the structure of the group of balanced labelings and related subgroups and quotient groups, including their algebraic properties.

Findings

Characterization of the group of balanced labelings

Analysis of subgroups and quotient groups related to balanced labelings

Self-contained discussion with references to graph connectivity algorithms

Abstract

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two closely related to it groups: the subgroup of balanced labelings which consists of functions vanishing on vertices and the corresponding factor-group. This work is completely self-contained, except the algorithm for obtaining the 3-edge-connected components of an undirected graph, for which we make appropriate references to the literature.