The property of kk-colourable graphs is uniquely decomposable

An additive hereditary graph property is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If P1, . . . ,Pn are graph properties, then a (P1, . . . ,Pn)-decomposition of a graph G is a partition E1, . . . , En of E(G) such that G[Ei], the subgraph of G induced by Ei, is in Pi, for i = 1, . . . , n. The sum of the properties P1, . . . ,Pn is the property P1 ⊕ · · · ⊕ Pn = {G ∈ I : G has a (P1, . . . ,Pn)-decomposition}. A property P is said to be decomposable if there exist non-trivial additive hereditary properties P1 and P2 such that P = P1 ⊕ P2. A property is uniquely decomposable if, apart from the order of the factors, it can be written as a sum of indecomposable properties in only one way. We show that not all properties are uniquely decomposable; however, the property of k-colourable graphs O is a uniquely decomposable property.