Proper edge-colorings with a rich neighbor requirement

Abstract

Under a given edge-coloring of a (multi)graph $G$, an edge is said to be rich if there is no color repetition among its neighboring edges; e.g., any isolated edge is rich. A rich-neighbor coloring of $G$ is a proper edge-coloring such that every non-isolated edge has at least one rich neighbor. For this weaker variant of strong edge-colorings, we believe that every connected subcubic graph apart form $K_4$ admits a rich-neighbor 5-coloring. In support of this, we show that every subcubic graph admits a rich-neighbor 7-coloring. The paper concludes with few open problems for subcubic graphs concerning the analogous notions of normal-neighbor colorings and poor-neighbor colorings.