Degree-truncated choosability of graphs

A graph G is called degree-truncated k-choosable if for every list assignment L with $\left|L\right(v\left)\right|\ge \min \left\{d_G\right(v),k\}$ for each vertex v, G is L-colourable. Richter asked whether every 3-connected non-complete planar graph is degree-truncated 6-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not degree-truncated 7-choosable. Then we prove that every 3-connected non-complete planar graph is degree-truncated 16-DP-colourable (and hence degree-truncated 16-choosable). We further prove that for an arbitrary proper minor closed family $G$ of graphs, let s be the minimum integer such that $K_{s,t}\notin G$ for some t, then there is a constant k such that every s-connected graph $G\in G$ other than a GDP tree is degree-truncated DP-k-colourable (and hence degree-truncated k-choosable), where a GDP-tree is a graph whose blocks are complete graphs or cycles. In particular, for any surface Σ, there is a constant k such that every 3-connected non-complete graph embeddable on Σ is degree-truncated DP-k-colourable (and hence degree-truncated k-choosable). The s-connectedness for graphs in $G$ (and 3-connectedness for graphs embeddable on Σ) is necessary, as for any positive integer k, $K_{s-1,k^{s-1}}\in G$ ($K_{2,k^2}$ is planar) is not degree-truncated k-choosable. Also, non-completeness is a necessary condition, as complete graphs are not degree-choosable.