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.