Variable degeneracy of graphs with restricted structures

Abstract

It is proved that strictly f -degenerate transversal is a generalization of list coloring, L-forested-coloring and DP-coloring. In this paper, we first give a result on strictly f -degenerate transversal of planar graphs without any configuration in Fig. 2. As a consequence, every planar graph without pairwise adjacent 3-, 4-, 5-and 6-cycles is DP-4-colorable, and has list vertex arboricity at most two. This result strengthens almost all the known results for planar graphs with DP-chromatic number four. The second main result is one on strictly f -degenerate transversal of toroidal graphs without any configuration in Fig. 3. As a consequence, every toroidal graph without pairwise adjacent 3-, 4-and 5-cycles is DP-4-colorable, and has list vertex arboricity at most two. In the final section, we present a result on strictly f -degenerate transversal of planar graphs without intersecting 5-cycles. As a consequence, every planar graph without intersecting 5-cycles is DP-4-colorable, and has list vertex arboricity at most two.