On 3-coloring of plane triangulations

For a 3-vertex coloring, a face of a triangulation whose vertices receive all three colors is called a vivid face with respect to it. In this paper, we show that for any triangulation G with n faces, there exists a coloring of G with at least 1/2n faces and construct an infinite series of plane triangulations such that any 3-coloring admits at most 1/5 (3n - 2) vivid faces.