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.
@article{bwmeta1.element.doi-10_7151_dmgt_1873,
author = {Izak Broere and Moroli D.V. Matsoha and Johannes Heidema},
title = {The Quest for A Characterization of Hom-Properties of Finite Character},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {479-500},
zbl = {1338.05189},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1873}
}
Izak Broere; Moroli D.V. Matsoha; Johannes Heidema. The Quest for A Characterization of Hom-Properties of Finite Character. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 479-500. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1873/