Characterization of cycle obstruction sets for improper coloring planar graphs^∗

For nonnegative integers k, d₁, . . ., d_k, a graph is (d₁, . . ., d_k)-colorable if its vertex set can be partitioned into k parts so that the ith part induces a graph with maximum degree at most d_i for all i ∈ {1, . . ., k}. A class C of graphs is balanced k-partitionable and unbalanced k-partitionable if there exists a nonnegative integer D such that all graphs in C are (D, . . ., D)colorable and (0, . . ., 0, D)-colorable, respectively, where the tuple has length k. A set X of cycles is a cycle obstruction set of a class C of planar graphs if every planar graph containing none of the cycles in X as a subgraph belongs to C. This paper characterizes all cycle obstruction sets of planar graphs to be balanced k-partitionable and unbalanced k-partitionable for all k; namely, we identify all inclusionwise minimal cycle obstruction sets for all k.