The Alon-Tarsi Number of Squares of Subcubic Planar Graphs without Cycles of Lengths $4$ to $8$

Abstract

The Alon--Tarsi number $AT(G)$ of a graph $G$, defined via the graph polynomial, is a strengthening of the list chromatic number ${\chi }_{\ell }(G)$. We study the Alon--Tarsi number of squares of planar graphs. The square of a graph $G$ is the graph obtained by joining every pair of vertices whose distance in $G$ is at most $2$. Recently, Kim and Luo (2026) proved that ${\chi }_{\ell }(G^2)\le 6$ for every subcubic planar graph containing no $k$-cycles for $4\le k\le 8$. We strengthen this result by proving that $AT(G^2)\le 6$ for every such graph $G$.