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Abstract  A sequences  1   ,   s  2   ,  ⋯  ,   s  k   ,   s  1   ,   s  2   ,  ⋯  ,   s  kis a repetition. A sequence  S  is nonrepetitive, if no subsequence of consecutive terms of  S  is a repetition. Let  G  be a plane graph. That is, a planar graph with a fixed embedding in the plane. A facial path consists of consecutive vertices on the boundary of a face. A facial nonrepetitive vertex coloring of a plane graph  G  is a vertex coloring such that the colors assigned to the vertices of any facial path form a nonrepetitive sequence. Letπ  f    (  G  )denote the minimum number of colors of a facial nonrepetitive vertex coloring of  G . Harant and Jendrol’ conjectured thatπ  f    (  G  )can be bounded from above by a constant. We prove thatπ  f    (  G  )   ≤  24    for any plane graph  G .

Facial Nonrepetitive Vertex Coloring of Plane Graphs