Locally irregular edge-coloring of claw-free graphs with maximum degree 4

A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular subgraph. A decomposable graph G is any graph which admits a locally irregular edge-coloring. The locally irregular chromatic index χ i r r ′ ( G ) of a decomposable graph G is the smallest number of colors required by a locally irregular edge-coloring of G . In this paper, we establish that χ i r r ′ ( G ) ≤ 5 for all claw-free graphs with maximum degree 4.