On maximum face-constrained coloring of plane graphs with no short face cycles

We prove that the vertices of each n-vertex plane graph G with minimum face cycle length g, g⩾5, can be colored using at leastg2−g−8g2+g−6 n+4g+4g2+g−6colors (for n⩾(g+3)/2) in such a way that G does not contain a polychromatic face, i.e., a face whose all the vertices have mutually different colors. In particular, if the girth of an n-vertex plane graph is at least five, then there is a coloring using at least ⌈n/2⌉+1 colors.