Facial edge ranking of plane graphs

Abstract

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