Determine whether every cubic graph is an AR-graph

Determine whether every cubic graph is an AR-graph, that is, whether every finite 3-regular simple graph admits an injective edge labeling by the integers from 1 through its number of edges such that every vertex has distinct subset sums of its incident edge labels.

Background

The paper introduces AR-labelings, in which each vertex has pairwise distinct sums for all subsets of its incident edge labels, and calls a graph an AR-graph when such a labeling uses exactly the labels 1 through the number of edges. It establishes that several families are AR-graphs, including paths, cycles, Hamiltonian cubic graphs, and cubic graphs whose vertex sets can be partitioned into vertex sets of cycles.

These results do not cover all cubic graphs. The concluding remarks explicitly identify the general cubic-graph case as unresolved, making it a direct open problem concerning the scope of the AR-graph class.