On the rich-neighbor edge-coloring of sparse graphs

Abstract

Let ϕ be a proper edge-coloring of a graph G. An edge e is rich if the edges adjacent to e receive distinct colors. In particular, a pendant edge and an isolated edge are both rich. Petruševski and Škrekovski (2024) introduced the concept of rich-neighbor edge-coloring as a weakening of strong edge-coloring. A proper k-edge-coloring ϕ is a rich-neighbor k-coloring if each non-isolated edge is adjacent to at least one rich edge. Petruševski and Škrekovski (2024) conjectured that every connected subcubic graph admits a rich-neighbor 5-coloring except for K4. We show that if G is a subcubic graph, then • [(1)] if [Formula presented], then G has a rich-neighbor 6-coloring; and • [(2)] if G is claw-free, then G has a rich-neighbor 6-coloring.