Facial Nonrepetitive Vertex Coloring of Plane Graphs

Abstract

A sequence s 1 , s 2 , . . . , s k , s 1 , s 2 , . . . , s k 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 π f (G) denote the minimum

On the Facial Thue Choice Index via Entropy Compression

“…The least number of colors required to construct such coloring is called the facial Thue chromatic index of G and denoted by π f (G). The theorem of Thue thus yields π f (P n ) ≤ 3 for all n. In [15] Havet et al proved that π f (G) ≤ 8 for every plane graph G, see [15,16,24] for other results (and [2,3,5,20] for further combinatorial extensions of the Thue's seminal result). This problem occurred much more demanding in the list setting.…”

“…In Havet et al. proved that for every plane graph G , see for other results (and for further combinatorial extensions of the Thue's seminal result). This problem occurred much more demanding in the list setting.…”

Planar graphs have bounded nonrepetitive chromatic number

“…The best known upper bound is O(log n) where n is the number of vertices, due to Dujmović, Frati, Joret, and Wood [16]. Note that several works have studied colourings of planar graphs in which only facial paths are required to be nonrepetitively coloured [4,8,33,34,44,45,48].…”

Fractional Thue chromatic number of graphs

“…They conjectured that the facial Thue chromatic numbers of plane graphs are bounded from above by a constant. This conjecture was proved by Barát and Czap [3].…”