The Quest for A Characterization of Hom-Properties of Finite Character

• Main Authors: Broere Izak, Matsoha Moroli D.V., Heidema Johannes
• Format: Article
• Language: English
• Published: Sciendo 2016-05-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: (countable) graph; homomorphism (of graphs); property of graphs; hom-property; (finitely-)induced-hereditary property; finitely determined property; (weakly) finite character; axiomatizable property; compactness theorems; core; connectedness; chromatic number; clique number; independence number; dominating set
• Online Access: https://doi.org/10.7151/dmgt.1873

A graph property is a set of (countable) graphs. A homomorphism from a graph G to a graph H is an edge-preserving map from the vertex set of G into the vertex set of H; if such a map exists, we write G → H. Given any graph H, the hom-property →H is the set of H-colourable graphs, i.e., the set of all graphs G satisfying G → H. A graph property P is of finite character if, whenever we have that F ∈ P for every finite induced subgraph F of a graph G, then we have that G ∈ P too. We explore some of the relationships of the property attribute of being of finite character to other property attributes such as being finitely-induced-hereditary, being finitely determined, and being axiomatizable. We study the hom-properties of finite character, and prove some necessary and some sufficient conditions on H for →H to be of finite character. A notable (but known) sufficient condition is that H is a finite graph, and our new model-theoretic proof of this compactness result extends from hom-properties to all axiomatizable properties. In our quest to find an intrinsic characterization of those H for which →H is of finite character, we find an example of an infinite connected graph with no finite core and chromatic number 3 but with hom-property not of finite character.