Colorings Of Plane Graphs With No Rainbow Faces

We prove that each n-vertex plane graph with girth g≥4 admits a vertex coloring with at least ⌈n/2⌉+1 colors with no rainbow face, i.e., a face in which all vertices receive distinct colors. This proves a conjecture of Ramamurthi and West. Moreover, we prove for plane graph with girth g≥5 that there is a vertex coloring with at least ${\left\lceil {\frac{{g - 3}} {{g - 2}}n - \frac{{g - 7}} {{2{\left( {g - 2} \right)}}}} \right\rceil }$ if g is odd and ${\left\lceil {\frac{{g - 3}} {{g - 2}}n - \frac{{g - 6}} {{2{\left( {g - 2} \right)}}}} \right\rceil }$ if g is even. The bounds are tight for all pairs of n and g with g≥4 and n≥5g/2−3.