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Facial Nonrepetitive Vertex Coloring of Plane Graphs


  • Contract grant sponsor: OTKA; Contract grant numbers: PD 75837; K 76099; Contract grant sponsor: János Bolyai Research Scholarship of the Hungarian Academy of Sciences; Contract grant sponsor: Australian Research Council; Contract grant number: DP120100197 (to J. B.).


A sequence math formula is 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. Let math formula denote the minimum number of colors of a facial nonrepetitive vertex coloring of G. Harant and Jendrol’ conjectured that math formula can be bounded from above by a constant. We prove that math formula for any plane graph G.