Hom-properties are uniquely factorizable into irreducible factors

Abstract

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 P 1, P 2,…, P n , a vertex ( P 1, P 2,…, P n) -partition of a graph G is a partition ( V 1, V 2,…, V n ) of V( G) such that each subgraph G[ V i ] induced by V i has property P i, i=1,2,…,n . The class of all vertex ( P 1, P 2,…, P n) -partitionable graphs is denoted by P 1∘ P 2∘⋯∘ P n . An additive hereditary property P is reducible if there exist additive hereditary properties P 1, P 2 such that P= P 1∘ P 2 , 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.