The Alon-Tarsi number of planar graphs without some forbidden configurations

The Alon-Tarsi number  of a graph G AT(G) defined as the smallest integer k admitting an Alon-Tarsi orientation with maximum out-degree at most k-1, satisfies the fundamental inequalities χl(G)≤AT(G)≤d(G)+1, where χl(G) and d(G) denote the list chromatic number and degeneracy respectively. Notably, for planar graphs without k-cycles where k∈{5,6}, the 3-degeneracy implies AT(G)≤4, and our main result extends this by proving AT(G)≤ 4 for planar graphs without 4-cycles adjacent to 3-cycles, thereby improving previous results from [JCTB 1999] and [Discrete Math. 2016].