Linear and 2-Frugal Choosability of Graphs of Small Maximum Average Degree

A proper vertex colouring of a graph G is 2-frugal (resp. linear) if the graph induced by the vertices of any two colour classes is of maximum degree 2 (resp. is a forest of paths). A graph G is 2-frugally (resp. linearly) L-colourable if for a given list assignment L:V(G)? 2^{\mathbb N}{L:V(G)\mapsto 2^{\mathbb N}} , there exists a 2-frugal (resp. linear) colouring c of G such that c(v) ? L(v){c(v) \in L(v)} for all v ? V(G){v\in V(G)} . If G is 2-frugally (resp. linearly) L-list colourable for any list assignment such that |L(v)| ≥ k for all v ? V(G){v\in V(G)}, then G is 2-frugally (resp. linearly) k-choosable. In this paper, we improve some bounds on the 2-frugal choosability and linear choosability of graphs with small maximum average degree.