Unique factorization of compositive hereditary graph properties

Abstract

Abstract. A graph property is any class of graphs that is closed under isomorphisms. A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs G 1 , G 2 ∈ P there exists a graph G ∈ P containing both G 1 and G 2 as subgraphs.Let H be any given graph on vertices v 1 , . . . , v n , n ≥ 2. A graph property P is Hfactorizable over the class of graph properties P if there exist P 1 , . . . , P n ∈ P such that P consists of all graphs whose vertex sets can be partitioned into n parts, possibly empty, satisfying:(1) for each i the graph induced by the i th non-empty partition part is in P i , and (2) for each i and j with i = j there are no edges between the i th and j th parts if v i and v j are non-adjacent vertices in H. If a graph property P is H-factorizable over P and we know the graph properties P 1 , . . . , P n , then we write P = H[P 1 , . . . , P n ]. In such a case, the presentation H[P 1 , . . . , P n ] is called a factorization of P over P. This concept generalizes graph homomorphisms and (P 1 , . . . , P n )-colorings.In this paper we investigate all H-factorizations of a graph property P over the class of all hereditary compositive graph properties for finite graphs H. It is shown that in many cases there is exactly one such factorization.