On the Facial Thue Choice Number of Plane Graphs Via Entropy Compression Method

Abstract

Let G be a plane graph. A vertex-colouring ϕ of G is called facial non-repetitive if for no sequence r 1 r 2 . . . r 2n , n ≥ 1, of consecutive vertex colours of any facial path it holds r i = r n+i for all i = 1, 2, . . . , n. A plane graph G is facial non-repetitively l-choosable if for every list assignment L : V → 2 N with minimum list size at least l there is a facial non-repetitive vertex-colouring ϕ with colours from the associated lists. The facial Thue choice number, π f l (G), of a plane graph G is the minimum number l such that G is facial non-repetitively l-choosable. We use the so-called entropy compression method to show that π f l (G) ≤ c∆ for some absolute constant c and G a plane graph with maximum degree ∆. Moreover, we give some better (constant) upper bounds on π f l (G) for special classes of plane graphs.

The list chromatic number of graphs with small clique number

“…Previous applications to graph colouring (e.g. [13,26,3,27,9,11,15]) involved situations where, throughout the algorithm, each vertex is guaranteed to have a large number of available colours to choose from. That is not true in this paper since the degree of a vertex can be much higher than its list-size.…”

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].…”

On a Generalization of Thue Sequences

“…For more results on graph colorings we refer the reader to an early survey of Grytczuk [15]. Additionally, a list version of this problem was studied for paths [18], trees [14], and plane graphs [23,24,26] where in the latter two the entropy compression method was used.…”