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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(...
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