List edge coloring of outer-1-planar graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number ${\chi }_l^{\mathrm{\prime }}(G)$ of any outer-1-planar graph $G$ with maximum degree $\mathrm{\Delta }(G)\ge 5$ is exactly its maximum degree. In this paper, we prove ${\chi }_l^{\mathrm{\prime }}(G)=\mathrm{\Delta }(G)$ for outer-1-planar graphs $G$ with $\mathrm{\Delta }(G)=4$ and with the crossing distance being at least 3.