Rainbowness of cubic plane graphs

The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G we prove that n 2 + ∗1 − 1 rb(G) n− ∗0 + 1, where ∗0 and ∗1 denote the independence number and the edge independence number, respectively, of the dual graph G∗ of G. We also prove that if the dual graph G∗ of an n-vertex cubic 3-connected plane graph G has a perfect matching then rb(G)= 3 4 n. © 2006 Elsevier B.V. All rights reserved. MSC: 05C15; 52B10