Petruševski–Škrekovski rich-neighbor 5-edge-coloring conjecture

Prove that every connected subcubic graph other than the complete graph K4 admits a rich-neighbor 5-edge-coloring.

Background

A proper edge-coloring of a graph is called rich-neighbor if every non-isolated edge is adjacent to an edge whose adjacent edges receive pairwise distinct colors. Petruševski and Škrekovski introduced this relaxation of strong edge-coloring and proved that every subcubic graph admits a rich-neighbor 7-edge-coloring.

The unresolved conjecture seeks to reduce the bound to five colors for every connected subcubic graph except K4. The paper proves the conjecture for subcubic graphs in which the vertices of degree three induce a graph of maximum degree at most one, thereby extending the previously known case where no two 3-vertices are adjacent. The full conjecture remains broader than the class treated by the paper.