Characterization of cubic graphs with balanced zero-neighborhood labelings

Characterize all cubic graphs that admit a balanced zero-neighborhood labeling, namely a binary vertex labeling with equally many vertices labeled 0 and 1 such that every vertex has neighborhood-weight zero modulo 2.

Background

The paper introduces balanced zero-neighborhood labelings as a binary labeling of a graph on 2n vertices in which exactly n vertices receive each label and the sum modulo 2 of the labels on the neighbors of every vertex is zero. These labelings arise naturally from projections of certain group distance magic labelings onto a [?]Z_2-coordinate.

The authors prove that generalized Petersen graphs GP(n,2) do not admit balanced zero-neighborhood labelings for n ≥ 3. They then explicitly leave open the broader classification problem for cubic graphs, seeking necessary and sufficient structural conditions for the existence of such labelings.