Variable degeneracy on toroidal graphs

DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization of list coloring and signed coloring. A new coloring, strictly $f$-degenerate transversal, is a further generalization of DP-coloring and $L$-forested-coloring. In this paper, we present some structural results on planar and toroidal graphs with forbidden configurations, and establish some sufficient conditions for the existence of strictly $f$-degenerate transversal based on these structural results. Consequently, (i) every toroidal graph without subgraphs isomorphic to the configurations in Fig.2 is DP-$4$-colorable, and has list vertex arboricity at most $2$, (ii) every toroidal graph without $4$-cycles is DP-$4$-colorable, and has list vertex arboricity at most $2$, (iii) every planar graph without subgraphs isomorphic to the configurations in Fig.3 is DP-$4$-colorable, and has list vertex arboricity at most $2$. These results improve upon previous results on DP-$4$-coloring [Discrete Math. 341~(7) (2018) 1983--1986; Bull. Malays. Math. Sci. Soc. 43~(3) (2020) 2271--2285] and (list) vertex arboricity [Discrete Math. 333 (2014) 101--105; Int. J. Math. Stat. 16~(1) (2015) 97--105; Iranian Math. Soc. 42~(5) (2016) 1293--1303].