Cover and variable degeneracy

Let f be a nonnegative integer valued function on the vertex set of graph. A graph is strictly f-degenerate if each nonempty subgraph ? has v such that deg?(v)<f(v). In this paper, we define new concept, transversal, which generalizes list coloring, signed DP-coloring, L-forested-coloring, and (f1,f2,…,fs)-partition. cover G H with V(H)=?v?V(G)Xv, where Xv={(v,1),(v,2),…,(v,s)}; edge M=?uv?E(G)Muv, Muv matching between Xu Xv. R?V(H) transversal |R?Xv|=1 for v?V(G). R H[R] f-degenerate. The main result paper degree type result, Brooks' theorem, Gallai's degree-choosable degree-colorable DP-degree-colorable result. We also give some structural results critical graphs respect to transversal. Using these results, can uniformly prove many known results. final section, pose open problems.