CHARACTERIZATION OF CYCLE OBSTRUCTION SETS FOR IMPROPER COLORING PLANAR GRAPHS

For nonnegative integers k, d(1),...,d(k), a graph is (d(1),...,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 is an element of{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.