Graphs maximal with respect to hom-properties

On universal graphs for hom-properties

“…The properties → H, H ∈ I, are called hom-properties or colour classes (see [14]). Graph homomorphisms and their structure were extensively investigated (see [9,12,13,17]), more references can be found in the survey [14] and in the book [10].…”

“…Let us recall some important properties of multiplications presented in [12,13]: Lemma 1. Let G :: be a multiplication of a graph G. If w, w * are two distinct vertices belonging to the same multivertex W of G :: , then there exists a homomorphism ψ :…”

“…For a finite graph H, the existence of a weakly universal graph W (H) in the class → H was in fact shown in [13]. In [1,2] A. Bonato gave an explicit construction of the universal (pseudo-homogeneous) graph M (H) in → H as a deterministic limit of a chain of finite H-colourable graphs.…”

Generalized Ramsey theory and decomposable properties of graphs

“…The following result will allow us to determine the chromatic number of the properties of the type I k 1 ⊕ I k 2 ⊕ • • • ⊕ I kn . In order to simplify the proof we introduce some concepts from [1] and [9].…”

A note on uniquely H-colorable graphs

“…We give an equivalent algebraic formulation of this condition in Theorem 3 (1). For related work on H-colourings and properties of cores, the reader is directed to [7,8].…”