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A facial unique-maximum coloring of a plane graph is a proper coloring of thevertices using positive integers such that each face has a unique vertex thatreceives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposeda strengthening of the Four Color Theorem conjecturing that all plane graphshave a facial unique-maximum coloring using four colors. This conjecture hasbeen disproven for general plane graphs and it was shown that five colorssuffice. In this paper we show that plane graphs, where vertices of degree atleast four induce a star forest, are facially unique-maximum 4-colorable. Thisimproves a previous result for subcubic plane graphs by Andova, Lidick\'y,Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing someproblems.

Facial unique-maximum colorings of plane graphs with restriction on big vertices