Hom-properties are uniquely factorizable into irreducible factors Original Research Article

For a simple graph H, a graph G is called H-colourable if there is a homomorphism from G to H (a mapping f : V(G)→V(H) such that uv∈E(G) implies f(u)f(v)∈E(H)). The class →H of H-colourable graphs is an additive hereditary property of graphs, called a hom-property. For hereditary properties P1,P2,…,Pn, a vertex (P1,P2,…,Pn)-partition of a graph G is a partition (V1,V2,…,Vn) of V(G) such that each subgraph G[Vi] induced by Vi has property Pi, i=1,2,…,n. The class of all vertex (P1,P2,…,Pn)-partitionable graphs is denoted by P1∘P2∘⋯∘Pn. An additive hereditary property P is reducible if there exist additive hereditary properties P1,P2 such that P=P1∘P2, it is irreducible otherwise. A graph is a core if it admits no homomorphism to any of its proper subgraphs. We prove that for any core H the hom-property →H is reducible if and only if H is a join (the Zykov sum of nonempty graphs). Moreover, we prove that the factorization of any hom-property →H into irreducible factors is unique.