Variable Degeneracy on Toroidal Graphs

DP-coloring was introduced by Dvořák 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 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 (Kim and Ozeki in Discrete Math 341(7):1983–1986. https://doi.org/10.1016/j.disc.2018.03.027, 2018; Sittitrai and Nakprasit in Bull Malays Math Sci Soc 43(3):2271–2285. https://doi.org/10.1007/s40840-019-00800-1, 2020) and (list) vertex arboricity (Choi and Zhang in Discrete Math 333:101–105. https://doi.org/10.1016/j.disc.2014.06.011, 2014; Huang et al. in Int J Math Stat 16(1):97–105, 2015; Zhang in Iranian Math Soc 42(5):1293–1303, 2016).