Facial edge ranking of plane graphs

A facial edge kk-ranking of a plane graph GG is a labeling of its edges with integers 1,…,k1,…,k such that every facial trail connecting two edges with the same label contains an edge with a greater label. The smallest integer kk such that GG has a facial edge kk-ranking is denoted by χfr′(G). We prove that χfr′(G)=O(logΔ∗) for 3-edge-connected plane graphs, where Δ∗Δ∗ is the maximum face size of GG.