The Alon-Tarsi number of planar graphs without cycles of lengths $4$ and $l$

Abstract: This paper proves that if $G$ is a planar graph without 4-cycles and $l$-cycles for some $l\in{5, 6, 7}$, then there exists a matching $M$ such that $AT(G-M)\leq 3$. This implies that every planar graph without 4-cycles and $l$-cycles for some $l\in{5, 6, 7}$ is 1-defective 3-paintable.