On the Rich-Neighbor Edge-Colorings of Subcubic Graphs

Abstract

Let be a proper edge-coloring of a graph G. An edge e is rich if all the edges adjacent to e receive distinct colors. Note that, if every non-isolated edge is rich, then the coloring is a strong edge-coloring, where every color class is an induced matching. Petruševski and Škrekovski (Discrete Math. 347:113803, 2024) introduced the concept of rich-neighbor edge-coloring as a weakening of strong edge-coloring. A proper k-edge-coloring is a rich-neighbork-coloring if each non-isolated edge is adjacent to at least one rich edge. Petruševski and Škrekovski (Discrete Math. 347:113803, 2024) conjectured that every connected subcubic graph admits a rich-neighbor 5-coloring except for. In this paper, we show that if G is a subcubic graph with no adjacent 3-vertices, then G has a rich-neighbor 5-coloring.