Hocquard–Lajou–Lužar conjecture on packing edge-colorings of cubic graphs

Establish that every cubic graph other than the Petersen graph and the Tietze graph admits a (1, 1, 1, 3)-packing edge-coloring.

Background

A (1, 1, 1, 3)-packing edge-coloring partitions the edges of a graph into three matching color classes and one color class whose distinct edges have edge-distance at least 4. Gastineau and Togni first conjectured that every cubic graph has such a coloring, but Hocquard et al. established that the Petersen and Tietze graphs are exceptions. The resulting conjecture asks whether these are the only exceptions.

The paper proves the conjecture for the restricted class of claw-free cubic graphs, including graphs with bridges, but does not settle the conjecture for all cubic graphs. Thus, the general cubic-graph case remains unresolved beyond the class addressed by Theorem 1.2.