On two problems of defective choosability of graphs

Abstract Given positive integers , and a nonnegative integer , we say a graph is ‐choosable if for every list assignment with for each and , there exists an ‐coloring of such that each monochromatic subgraph has maximum degree at most . In particular, ‐choosable means ‐colorable, ‐choosable means ‐choosable and ‐choosable means ‐defective ‐choosable. This paper proves that there are 1‐defective 3‐choosable planar graphs that are not 4‐choosable, and for any positive integers , and nonnegative integer , there are ‐choosable graphs that are not ‐choosable. These results answer questions asked by Wang and Xu, and Kang, respectively. Our construction of ‐choosable but not ‐choosable graphs generalizes the construction of Král' and Sgall for the case .