Defective DP-colorings of sparse multigraphs

Decomposing planar graphs into graphs with degree restrictions

“…A graph $$G$$ is $$h$$‐ defective $$k$$‐ choosable if for every $$k$$‐list assignment $$L$$ of $$G$$, there is an $$L$$‐coloring of $$G$$ in which each vertex $$v$$ has at most $$h$$ neighbors colored the same color as $$v$$. The concept of $$h$$‐ defective $$k$$‐ paintability is an online version of $$h$$‐defective $$k$$‐choosability, defined through a two‐person game (see [9] for its definition), and $$h$$‐ defective $$k$$‐ DP‐colourability is a generalization of $$h$$‐defective $$k$$‐choosability (see [11] for its definition). We remark that $$(d,h)$$…”

“…A graph G is h-defective k-choosable if for every k-list assignment L of G, there is an L-coloring of G in which each vertex v has at most h neighbors colored the same color as v. The concept of h-defective k-paintability is an online version of h-defective k-choosability, defined through a two-person game (see [9] for its definition), and h-defective k-DP-colourability is a generalization of h-defective k-choosability (see [11] for its definition). We remark that d h ( , )decomposable graphs are easily seen to be h-defective d ( + 1)-choosable, h-defective d ( + 1)paintable, as well as h-defective d ( + 1)-DP-colorable.…”