Formula omitted]-free graphs with no odd holes

Abstract

All K 4 -free graphs with no odd hole and no odd antihole are three-colourable, but what about K 4 -free graphs with no odd hole? They are not necessarily three-colourable, but we prove a conjecture of Ding that they are all four-colourable. This is a consequence of a decomposition theorem for such graphs; we prove that every such graph either has no odd antihole, or belongs to one of two explicitly-constructed classes, or admits a decomposition.