NOTE Rainbowness of cubic polyhedral 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 have different colours. For a cubic polyhedral (i.e. 3-connected plane) graph G we prove that n 2 + α∗ 1 − 1 ≤ rb(G) ≤ n− α∗ 0 + 1 , where α∗ 0 and α ∗ 1 denote the independence number and the edge independence number, respectively, of the dual graph G∗ of G. Moreover, we show that the lower bound is tight and that the upper bound cannot be less than n− α∗ 0 in general. We also prove that if the dual graph G∗ of an n-vertex cubic polyhedral graph G has a perfect matching then