Facial Rainbow Edge-Coloring of Plane Graphs

An edge-coloring of a loopless plane graph G is a facial rainbow edge-coloring if any two edges of G contained in the same facial path have distinct colors. The facial rainbow edge-number of a graph G , denoted erb ( G ) , is the minimum number of colors that are necessary in any facial rainbow edge-coloring. In the present note we prove that erb ( G ) ≤ ⌊ 3 2 ( L ( G ) + 1 ) ⌋ for all connected loopless plane graphs, where L ( G ) is the length of the longest facial path of G . This bound is tight. For the family of all 3-connected plane graphs this bound is improved to L ( G ) + 2 . For trees there is erb ( G ) ≤ ⌊ 3 2 L ( G ) ⌋ which is also tight. Moreover, if G is a tree with L ( G ) ≥ 7 and without degree two vertices, then erb ( G ) = L ( G ) .