Toroidal graphs without $K_{5}^{-}$ and 6-cycles

Cai et al.\ proved that a toroidal graph $G$ without $6$-cycles is $5$-choosable, and proposed the conjecture that $\textsf{ch}(G) = 5$ if and only if $G$ contains a $K_{5}$ [J. Graph Theory 65 (2010) 1--15], where $\textsf{ch}(G)$ is the choice number of $G$. However, Choi later disproved this conjecture, and proved that toroidal graphs without $K_{5}^{-}$ (a $K_{5}$ missing one edge) and $6$-cycles are $4$-choosable [J. Graph Theory 85 (2017) 172--186]. In this paper, we provide a structural description, for toroidal graphs without $K_{5}^{-}$ and $6$-cycles. Using this structural description, we strengthen Choi's result in two ways: (I) we prove that such graphs have weak degeneracy at most three (nearly $3$-degenerate), and hence their DP-paint numbers and DP-chromatic numbers are at most four; (II) we prove that such graphs have Alon-Tarsi numbers at most $4$. Furthermore, all of our results are sharp in some sense.