On the facial Thue choice number of plane graphs via entropy compression method

The Thue choice number versus the Thue chromatic number of graphs

“…the Thue chromatic index, π ′ (G) -see [11], [30]; the Thue choice index, π ′ l (G) -see [26]; the facial Thue chromatic index, π ′ f (G) -see [26], [38], [39], [62]; the facial Thue choice index, π ′ f l (G) -see [26], [55], [59]), while the Thue graph parameter connected with nonrepetitive vertex colourings or total colourings will be called number and abbreviated without apostrophe (e.g. the Thue chromatic number, π(G) -see [11], [12], [15], [26], [27], [28], [29], [30], [34], [35], [40], [41], [53], [54], [58]; the Thue choice number, π l (G) -see [24], [26], [32] 3 , [42]; the facial Thue choice number, π f l (G) -see [26], [56]; the facial Thue chromatic number, π f (G) -see [5], [26], [33], [34], unhapilly, with the same abbreviation like the fractional Thue chromatic number, π f (G) -see [40], [64]; for the Thue parameters related to total Thue colourings see [43], [60]). We will also follow this idea and use the abbreviation π(G) for the Thue chromatic number and π l (G) for the Thue choice number of a graph G. Except of the few notation defined throughout the paper we will use the standard terminology according to Bondy and Murty [10].…”

“…Dujmović et al [21] using the entropy compression method (see e.g. [26], [31], [48], [49], [55], [56]) also improved the constant c in the upper bound c∆ 2 and showed that for large graphs c even tends to 1: Theorem 6. (Dujmović, Joret, Kozik, Wood, 2015+, [21]) For every graph G with maximum degree ∆ > 1,…”