Rainbow faces in edge-colored plane graphs

A face of an edge colored plane graph is called rainbow if all its edges receive distinct colors. The maximum number of colors used in an edge coloring of a connected plane graph G with no rainbow face is called the edge-rainbowness of G. In this paper we prove that the edge-rainbowness of G equals to the maximum number of edges of a connected bridge face factor H of G, where a bridge face factor H of a plane graph G is a spanning subgraph H of G in which every face is incident with a bridge and the interior of any one face f ∈ F (G) is a subset of the interior of some face f ′ ∈ F (H). We also show upper and lower bounds on the edge-rainbowness of graphs based on edge connectivity, girth of the dual G∗ and other basic graph invariants. Moreover, we present infinite classes of graphs where these equalities are attained.