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

Abstract

Let G be a plane graph. A vertex-colouring $\varphi$ of G is called facial non-repetitive if for no sequence $r_1 r_2 \ldots r_{2n}$, $n\ge 1$, of consecutive vertex colours of any facial path it holds $r_i=r_{n+i}$ for all $i=1,2,\ldots ,n$. A plane graph G is facial non-repetitively k -choosable if for every list assignment $L:V\rightarrow 2^{\mathbb {N}}$ with minimum list size at least k there is a facial non-repetitive vertex-colouring $\varphi$ with colours from the associated lists. The facial Thue choice number, $\pi _{fl}(G)$, of a plane graph G is the minimum number k such that G is facial non-repetitively k-choosable. We use the so-called entropy compression method to show that $\pi _{fl} (G)\le c \varDelta$ for some absolute constant c and G a plane graph with maximum degree $\varDelta$. Moreover, we give some better (constant) upper bounds on $\pi _{fl} (G)$ for special classes of plane graphs.