Critically paintable, choosable or colorable graphs

Abstract

We extend results about critically k-colorable graphs to choosability and paintability (list colorability and on-line list colorability). Using a strong version of Brooks’ Theorem, we generalize Gallai’s Theorem about the structure of the low-degree subgraph of critically k-colorable graphs, and introduce a more adequate lowest-degree subgraph. We prove lower bounds for the edge density of critical graphs, and generalize Heawood’s Map-Coloring Theorem about graphs on higher surfaces to paintability. We also show that on a fixed given surface, there are only finitely many critically k-paintable/k-choosable/k-colorable graphs, if k≥6. In this situation, we can determine in polynomial time k-paintability, k-choosability and k-colorability, by giving a polynomial time coloring strategy for “Mrs. Correct”. Our generalizations of k-choosability theorems also concern the treatment of non-constant list sizes (non-constant k). Finally, we use a Ramsey-type lemma to deduce all 2-paintable, 2-choosable, critically 3-paintable and critically 3-choosable graphs, with respect to vertex deletion and to edge deletion.