A group-labeled graph is a graph whose vertices and edges have been assigned labels from some abelian group. The weight of a subgraph of a group-labeled graph is the sum of the labels of the vertices and edges in the subgraph. A group-labeled graph is said to be balanced if the weight of every cycle in the graph is zero. We give a characterization of balanced group-labeled graphs that generalizes the known characterizations of balanced signed graphs and consistent marked graphs. We count the number of distinct balanced labellings of a graph by a finite abelian group Γ and show that this number depends only on the order of Γ and not its structure. We show that all balanced labellings of a graph can be obtained from the all-zero labeling using simple operations. Finally, we give a new constructive characterization of consistent marked graphs and markable graphs, that is, graphs that have a consistent marking with at least one negative vertex. © 2011 Elsevier B.V. All rights reserved.
Joglekar, M., Shah, N., & Diwan, A. A. (2012). Balanced group-labeled graphs. Discrete Mathematics, 312(9), 1542–1549. https://doi.org/10.1016/j.disc.2011.09.021