FACIAL RAINBOW EDGE-COLORING OF SIMPLE 3-CONNECTED PLANE GRAPHS

A facial rainbow edge-coloring of a plane graph G is an edge-coloring such that any two edges receive distinct colors if they lie on a common facial path of G. The minimum number of colors used in such a coloring is denoted by erb(G). Trivially, erb(G) >= L(G) + 1 holds for every plane graph without cut-vertices, where L(G) denotes the length of a longest facial path in G. Jendrof in 2018 proved that every simple 3-connected plane graph admits a facial rainbow edge-coloring with at most L(G) + 2 colors, moreover, this bound is tight for L(G) = 3. He also proved that erb(G) = L(G) +1 for L(G) {3, 4, 5} . He posed the following conjecture: There is a simple 3-connected plane graph G with L(G) = 4 and erb(G) = L(G)+2. In this note we answer the conjecture in the affirmative.