Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties
Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if and are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.