Let G be a connected plane graph that can have loops and multiple edges. An l-facial edge-coloring of a plane graph G is a coloring of edges of G such that any two edges, that share the same facial trail of length at most \(l + 1\) , receive distinct colors. It is an edge variant of the l-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It was conjectured by Lužar et al. in 2015 that every plane graph admits an l-facial edge-coloring with at most \(3l + 1\) colors for any \(l \ge 1\) . It is known that the bound \(3l+1\) is tight for general plane graphs. The conjecture was recently confirmed for \(l \le 3\) by Horňák, Lužar and Štorgel (3-facial edge-coloring of plane graphs, Discrete Math. 346 (2023) 113312). In this note we prove that the conjecture holds, in the case when \(l \ge 4\) , for every graph whose reduction (the graph obtained from G by suppressing all its 2-vertices) is 3-edge connected, and the length of the longest path in G with interior vertices of degree 2 is at most \(\frac{3l + 1}{10}\) .
